local system on $C$. Nontrivial local system will appear in Section

A shift is required to make the isomorphism (<ref>) preserve
cohomological dimension and the action of Frobenius. Indeed, we have

\[
\deg \be\of{\alpha} = \deg \be\of{\bpt}-1\,, \quad
\textup{weight}\, \be\of{\alpha} = \textup{weight} \,
\be\of{\bpt}-\tfrac12\,,
\]

where the degree is the cohomological degree and weight refers to the
absolute values of the Frobenius eigenvalues. So, if we want to get
these operators as $H^1$, we need both a shift in cohomology and a
Tate twist. We define

\begin{equation}
\label{eq:42}
H\Omega_0 =  H^1(C,\Omega^1(\bX^\vee_0)))\sTate{2}\,,
\end{equation}

\[
\sTate{2}=[2]\Tate{1}
\]

denotes the pure combination of the cohomological shift and
Tate twist, and
where $\Omega^1$ was originally placed in degree $0$, compare with
From the Poisson structure on the cone and multiplication in cohomology, we get a symmetric

\begin{equation}
\label{eq:41}
S^2 H\Omega_0
\xrightarrow{\quad  \cup \otimes \{ \, \cdot \, , \, \cdot \, \}
\quad}
H^2(C,\cO_{\bX^\vee_0}) \sTate{2}=\cO_{\bX^\vee_0}= \cA^2 \,.
\end{equation}

We define

\[
\Cliff_{\bX^\vee_0} = \textup{Clifford}(H\Omega_0) \,.
\]

Formulas (<ref>) and (<ref>), and the naturality of
all operations, give the following

We have a degree-preserving
isomorphism $ \cA^{\ge 1}(C)\cong \Cliff_{\bX^\vee_0} $. Moreover,
this isomorphism is Frobenius-equivariant.

We define

\begin{equation}
\label{eq:42*2}
\sigma: \bX^\vee \cong T^* \bP^1 \to \bX^\vee_0
\end{equation}

as the blow-up at the vertex of the cone. This is the world's most
basic equivariant symplectic resolution. As above, $\bX^\vee$
gets its own sheaf of Clifford algebras $\Cliff_{\bX^\vee}$ with
a homomorphism

\[
\sigma^*: \Cliff_{\bX^\vee_0}  \to \Cliff_{\bX^\vee} \,.
\]

There is thus a well-define pushforward $\sigma_*$ of Clifford modules.

In modern representation theory, understanding the representation of
quantizations of Poisson varieties via (derived) pullbacks and
pushforwards under maps like $\sigma$ is a very important technical
tool known as localization. Our situation is simpler in that
we deal with sheaves of Clifford algebras instead of really
noncommutative algebras like the full $\cA(C)$. In the end,
however, the full structure of the $\cA(C)$-module is uniquely

We will show, and this will be the main result of this section, that
the module $\Hd(\QMN)$ is the pushforward of a suitable twist of the spinor bundle
$\Spin_{\bX^\vee}$. This will be, however, not the
ordinary pushforward, but rather pushforward with support in
a certain Lagrangian $\bfL \subset \bX^\vee$.

It will be convenient to visualize various geometric notions
using the toric diagram of $\bX^\vee=T^*\bP^1$ as seen on the right in Figure
(<ref>) below. It is also
known as the Newton or moment polyhedron of $T^*\bP^1$. Its two vertices
correspond to the fixed points $0,\infty \in \bP^1$. Edges
correspond to the 1-dimensional torus orbits.

Canonically, diagrams like (<ref>) lie in the
dual of the Lie algebra of a maximal torus

\[
\{(a,q)\} \subset \Aut(T^*\bP^1)
\]

where $a$ acts by $\diag(a^{-1},a) \in SL(2)$ and $q$ scales the
cotangent fibers with weight $q^{-1}$. Lattice points in toric
diagram are the weights of sections of line
bundles. Specifically, in  (<ref>) we see the
weights $H^0(\cO_{\bX^\vee})$ and
$H^0(\cO_{\bX^\vee}(8))$. Here $\cO_{\bX^\vee}(8)$ is a random ample
line bundle. For the natural linearization, the weights of the corner
vertices for $\cO_{\bX^\vee}(8)$ are $a^{8}$ and $a^{-8}$,
respectively. In Figure <ref> we indicated the weights of
neighboring vertices relative to the weights of the corner vertices.

\begin{equation}
\label{eq:301}
\raisebox{-1.5cm}{
\begin{tikzpicture}[scale=1]
\draw [line width=1pt] (-8.2,2.2)--(-6,0)--(-3.8,2.2);
\fill [fill=gray!20] (-8.2,2.2)--(-6,0)--(-3.8,2.2)--(-8.2,2.2);
%\node (0) at (-2.3,1.7) {$\bff\of{\bpt}$};
%\node (1) at (6.3,1.7) {$\be\of{\bpt}$};
\node at (-6,0) {${\scriptstyle \bullet}$};
\node at (-13/2,1/2) {${\scriptstyle \bullet}$};
\node at (-6,1/2) {${\star}$};
\node at (-11/2,1/2) {${\scriptstyle \bullet}$};
\node at (-7,1) {${\scriptstyle \bullet}$};
\node at (-13/2,1) {${\scriptstyle \bullet}$};
\node at (-6,1) {${\scriptstyle \bullet}$};
\node at (-11/2,1) {${\scriptstyle \bullet}$};
\node at (-5,1) {${\scriptstyle \bullet}$};
\node at (-15/2,3/2) {${\scriptstyle \bullet}$};
\node at (-7,3/2) {${\scriptstyle \bullet}$};
\node at (-13/2,3/2) {${\scriptstyle \bullet}$};
\node at (-6,3/2) {${\scriptstyle \bullet}$};
\node at (-11/2,3/2) {${\scriptstyle \bullet}$};
\node at (-5,3/2) {${\scriptstyle \bullet}$};
\node at (-9/2,3/2) {${\scriptstyle \bullet}$};
\node at (-8,2) {${\scriptstyle \bullet}$};
\node at (-15/2,2) {${\scriptstyle \bullet}$};
\node at (-7,2) {${\scriptstyle \bullet}$};
\node at (-13/2,2) {${\scriptstyle \bullet}$};
\node at (-6,2) {${\scriptstyle \bullet}$};
\node at (-11/2,2) {${\scriptstyle \bullet}$};
\node at (-5,2) {${\scriptstyle \bullet}$};
\node at (-9/2,2) {${\scriptstyle \bullet}$};
\node at (-4,2) {${\scriptstyle \bullet}$};
\node[rectangle,anchor=east, inner sep=0] (2) at (-6.7,0.5)  {$\bff\of{\bpt}$};
\node[rectangle,anchor=west,inner sep=0] (3) at (-5.3,0.5) {$\be\of{\bpt}$};
%\node[rectangle,anchor=south,inner sep=0] (3) at (-6,0.55){\scalebox{0.7}{$\bh\!\of{\bpt}$}};
\draw [line width=1pt] (-2.2,2.2)--(0,0)--(4,0)--(6.2,2.2);
\fill [fill=gray!20] (-2.2,2.2)--(0,0)--(4,0)--(6.2,2.2)--(-2.2,2.2);
% \node (0) at (-2.3,1.7) {$\bff\of{\bpt}$};
%\node (1) at (6.3,1.7) {$\be\of{\bpt}$};
\node[rectangle,anchor=east, inner sep=0] (2) at (-0.73,0.5)  {$qa^{2}$};
\node[rectangle,anchor=west,inner sep=0] (3) at (4.7,0.5) {$qa^{-2}$};
\node[rectangle,anchor=north,inner sep=0] (4) at (0.55,-0.1)
\node[rectangle,anchor=north,inner sep=0] (5) at (3.5,-0.1)
\node[rectangle,anchor=north east,inner sep=1] (6) at (0,0)
\node[rectangle,anchor=north west,inner sep=1] (7) at (4,0)
%\node (8) at (0,0) {${\scriptscriptstyle \bullet}$};
%\node (8) at (4,0) {${\scriptscriptstyle \bullet}$};

\draw[ultra thick, ->] (-3,0.7) arc (-60:-300:0.5);
%\node at (-3,1) {\scalebox{\rotatebox{-90}{$\circlearrowright $}};

\node at (0,0) {${\scriptstyle \bullet}$};
\node at (1/2,0) {${\scriptstyle \bullet}$};
\node at (1,0) {${\scriptstyle \bullet}$};
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\node at (0,1/2) {${\scriptstyle \bullet}$};
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\node at (5/2,1) {${\scriptstyle \bullet}$};
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\node at (4,1) {${\scriptstyle \bullet}$};
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\node at (5,1) {${\scriptstyle \bullet}$};
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\node at (0,3/2) {${\scriptstyle \bullet}$};
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\node at (5/2,3/2) {${\scriptstyle \bullet}$};
\node at (3,3/2) {${\scriptstyle \bullet}$};
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\node at (4,3/2) {${\scriptstyle \bullet}$};
\node at (9/2,3/2) {${\scriptstyle \bullet}$};
\node at (5,3/2) {${\scriptstyle \bullet}$};
\node at (11/2,3/2) {${\scriptstyle \bullet}$};
\node at (-2,2) {${\scriptstyle \bullet}$};
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\node at (-1,2) {${\scriptstyle \bullet}$};
\node at (-1/2,2) {${\scriptstyle \bullet}$};
\node at (0,2) {${\scriptstyle \bullet}$};
\node at (1/2,2) {${\scriptstyle \bullet}$};
\node at (1,2) {${\scriptstyle \bullet}$};
\node at (3/2,2) {${\scriptstyle \bullet}$};
\node at (2,2) {${\scriptstyle \bullet}$};
\node at (5/2,2) {${\scriptstyle \bullet}$};
\node at (3,2) {${\scriptstyle \bullet}$};
\node at (7/2,2) {${\scriptstyle \bullet}$};
\node at (4,2) {${\scriptstyle \bullet}$};
\node at (9/2,2) {${\scriptstyle \bullet}$};
\node at (5,2) {${\scriptstyle \bullet}$};
\node at (11/2,2) {${\scriptstyle \bullet}$};
\node at (6,2) {${\scriptstyle \bullet}$};
\end{tikzpicture}}
\end{equation}

Note that $H^0(\cO_{\bX^\vee})=\cA^2$. We have marked the
weights of $\bff\of{\bpt}$ and $\be\of{\bpt}$, while
the weight of $\bh\of{\bpt}$ is marked by a star $\star$.
$H^0(\cO_{\bX^\vee}(8))$ is a module over $H^0(\cO_{\bX^\vee})$,
the left
part of (<ref>) acts on the right part by addition, as
indicated by the curved arrow.

As customary in toric geometry, one may view the shapes in Figure
(<ref>) as a combinatorial equivalent of the geometric shapes
of the cone $\bX^\vee_0$ and its resolution $\bX^\vee$, respectively.

We define

\begin{align}
\label{eq:328}
\bfL &= \{ \bff\of{\bpt} = 0 \}  \\
\label{eq:336}
&= \{ x \in \bX^\vee, \exists \lim_{a\to 0} a\cdot x \} \,.
\end{align}

This is the union of two attracting manifolds, namely

_0    = { x ∈^∨, lim_a→0 a·x =
0}      =  ,

_∞   = { x ∈^∨, lim_a→0 a·x =
∞}      =  ,

see Figure (<ref>).

To get a feeling for what the pushforward with support in $\bfL$ looks
like, let us compute it for $\cO_{\bX^\vee}(8)$, which means we
consider the local homology groups $\Hd_{\bfL}(\cO_{\bX^\vee}(8))$.
The exact sequence of a pair for $\bfL_\infty \subset \bfL$ expresses
$\Hd_{\bfL}$ in terms of $\Hd_{\bfL_\infty}$ and
$\Hd_{\bfL_0}$. Both of these can be computed in a
single affine chart. This gives

\[
H^i_{\bfL_\infty}(\cO(8))= H^i_{\bfL_0}(\cO(8))=0 \,, \quad i \ne 1 \,,
\]

\begin{equation}
\label{eq:329}
\xymatrix{
0 \ar[r] & H^1_{\bfL_\infty}(\cO(8))  \ar@{=}[d] \ar[r] & H^1_{\bfL}(\cO(8))
\ar[r] & H^1_{\bfL_0}(\cO(8)) \ar[r]
\ar@{=}[d] & 0\,,\\
a^{8} (\be \bh^{-1})^{8}  \, R[\be, (\bff \bh^{-1})^{\pm 1}]/ R[\be , \bff \bh^{-1}]
a^{8} R[\be \bh^{-1}, \bff^{\pm 1}]/ R[\be \bh^{-1}, \bff]
\end{equation}

where $R$ is base ring of coefficients. See the
picture of the corresponding modules in Figure
(<ref>), where the submodule is shaded in a darker shade of
gray. The middle term (<ref>) is the unique
nontrivial extension between them.

\begin{equation}
\label{eq:301-2}
\raisebox{-1cm}{\begin{tikzpicture}[scale=1]
\draw [line width=0.5pt] (-2.2,2.2)--(0,0);
\draw [line width=2pt] (0,0)--(3.85,0);
\draw [line width=2pt] (4,0)--(6.2,2.2);
\node at (0,0) {${\scriptstyle \bullet}$};
\node at (4,0) {${\scriptstyle \bullet}$};
%\draw [line width=0.5pt] (2,-2)--(0.5,-0.5)--(7.5,-0.5);
%\draw [line width=0.5pt] (6.7,2.2)--(4.5,0)--(7.7,0);
%\fill [fill=peach1!70]
\fill [fill=gray!40]
% \fill [fill=blue1!70]
\fill [fill=gray!20]
% \fill [fill=gray!20] (-2,2)--(0,0)--(4,0)--(6,2)--(-2,2);

\node[rectangle,anchor=south, inner sep=0] at (2.1,0.1)
\node[rectangle,anchor=south east, inner sep=0] at (5,1.1) {$\bfL_\infty$};

% \node (0) at (-2.3,1.7) {$\bff\of{\bpt}$};
%  \node (1) at (6.3,1.7) {$\be\of{\bpt}$};
%  \node[rectangle,anchor=south west, inner sep=0] (2) at (-0.7,0.7)  {$\chi^2_\rho(1)$};
%  \node[rectangle,anchor=south east,inner sep=0] (3) at (4.7,0.7) {$\chi^{-2}_\rho(1)$};
% \node[rectangle,anchor=south,inner sep=0] (4) at (1.1,0.1)
% {$\chi^{-2}_\rho$};
% \node[rectangle,anchor=south,inner sep=0] (5) at (3,0.1)
% {$\chi^{2}_\rho$};
% \node[rectangle,anchor=north east,inner sep=1] (6) at (0,0)
% {$0$};
%  \node[rectangle,anchor=north west,inner sep=1] (7) at (4,0)
%  {$\infty$};
%  \node (8) at (0,0) {${\scriptscriptstyle \bullet}$};
%  \node (8) at (4,0) {${\scriptscriptstyle \bullet}$};
\node at (1/2,-1/2) {${\scriptstyle \bullet}$};
\node at (1,-1/2) {${\scriptstyle \bullet}$};
\node at (3/2,-1/2) {${\scriptstyle \bullet}$};
\node at (2,-1/2) {${\scriptstyle \bullet}$};
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\node at (3,-1/2) {${\scriptstyle \bullet}$};
\node at (7/2,-1/2) {${\scriptstyle \bullet}$};
\node at (4,-1/2) {${\scriptstyle \bullet}$};
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\node at (5,-1/2) {${\scriptstyle \bullet}$};
\node at (11/2,-1/2) {${\scriptstyle \bullet}$};
\node at (6,-1/2) {${\scriptstyle \bullet}$};
\node at (13/2,-1/2) {${\scriptstyle \bullet}$};
\node at (7,-1/2) {${\scriptstyle \bullet}$};
\node at (15/2,-1/2) {${\scriptstyle \bullet}$};
\node at (1,-1) {${\scriptstyle \bullet}$};
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\node at (2,-1) {${\scriptstyle \bullet}$};
\node at (5/2,-1) {${\scriptstyle \bullet}$};
\node at (3,-1) {${\scriptstyle \bullet}$};
\node at (7/2,-1) {${\scriptstyle \bullet}$};
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\node at (9/2,-1) {${\scriptstyle \bullet}$};
\node at (5,-1) {${\scriptstyle \bullet}$};
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\node at (15/2,-1) {${\scriptstyle \bullet}$};
\node at (3/2,-3/2) {${\scriptstyle \bullet}$};
\node at (2,-3/2) {${\scriptstyle \bullet}$};
\node at (5/2,-3/2) {${\scriptstyle \bullet}$};
\node at (3,-3/2) {${\scriptstyle \bullet}$};
\node at (7/2,-3/2) {${\scriptstyle \bullet}$};
\node at (4,-3/2) {${\scriptstyle \bullet}$};
\node at (9/2,-3/2) {${\scriptstyle \bullet}$};
\node at (5,-3/2) {${\scriptstyle \bullet}$};
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\node at (6,-3/2) {${\scriptstyle \bullet}$};
\node at (13/2,-3/2) {${\scriptstyle \bullet}$};
\node at (7,-3/2) {${\scriptstyle \bullet}$};
\node at (15/2,-3/2) {${\scriptstyle \bullet}$};
\node at (2,-2) {${\scriptstyle \bullet}$};
\node at (5/2,-2) {${\scriptstyle \bullet}$};
\node at (3,-2) {${\scriptstyle \bullet}$};
\node at (7/2,-2) {${\scriptstyle \bullet}$};
\node at (4,-2) {${\scriptstyle \bullet}$};
\node at (9/2,-2) {${\scriptstyle \bullet}$};
\node at (5,-2) {${\scriptstyle \bullet}$};
\node at (11/2,-2) {${\scriptstyle \bullet}$};
\node at (6,-2) {${\scriptstyle \bullet}$};
\node at (13/2,-2) {${\scriptstyle \bullet}$};
\node at (7,-2) {${\scriptstyle \bullet}$};
\node at (15/2,-2) {${\scriptstyle \bullet}$};
\node at (9/2,0) {${\scriptstyle \bullet}$};
\node at (5,0) {${\scriptstyle \bullet}$};
\node at (11/2,0) {${\scriptstyle \bullet}$};
\node at (6,0) {${\scriptstyle \bullet}$};
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\node at (15/2,0) {${\scriptstyle \bullet}$};
\node at (5,1/2) {${\scriptstyle \bullet}$};
\node at (11/2,1/2) {${\scriptstyle \bullet}$};
\node at (6,1/2) {${\scriptstyle \bullet}$};
\node at (13/2,1/2) {${\scriptstyle \bullet}$};
\node at (7,1/2) {${\scriptstyle \bullet}$};
\node at (15/2,1/2) {${\scriptstyle \bullet}$};
\node at (11/2,1) {${\scriptstyle \bullet}$};
\node at (6,1) {${\scriptstyle \bullet}$};
\node at (13/2,1) {${\scriptstyle \bullet}$};
\node at (7,1) {${\scriptstyle \bullet}$};
\node at (15/2,1) {${\scriptstyle \bullet}$};
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\node at (13/2,3/2) {${\scriptstyle \bullet}$};
\node at (7,3/2) {${\scriptstyle \bullet}$};
\node at (15/2,3/2) {${\scriptstyle \bullet}$};
\node at (13/2,2) {${\scriptstyle \bullet}$};
\node at (7,2) {${\scriptstyle \bullet}$};
\node at (15/2,2) {${\scriptstyle \bullet}$};
\end{tikzpicture}}
\end{equation}

In (<ref>), we picked a specific linearization of
$\cO(8)$. Choosing a different linearization translates the whole
picture (<ref>), while replacing $\cO(8)$ by a general $\cO(m)$
makes the two shaded pieces slide relative each other.

The diagram (<ref>) visualizes the structure of
$\Hd_{\bfL}(\cO_{\bX^\vee}(m))$ as a $\cA^2$-module. In particular,
$\bff$ and $\bh$ act locally nilpotently, while $\be$-torsion is
finite-dimensional. Evidently, one can also describe the result of
the computation as follows

\[
H^1_{\bfL}(\cO(8)) = H^0_{\bfL}(\cO(8))[\bff^{-1}] \big/
H^0_{\bfL}(\cO(8)) \,.
\]

As a result of taking the cohomology with support in the
attracting manifold $\bfL$, the $a$-grading of all resulting modules
is bounded from above. In the Eisenstein series context, the variable $a$ weights
a quasisection $s$ by $a^{-\deg(s)}$. Since $\deg(s)$ is bounded below
for fixed $\cM$, it makes sense to look for a description of the
$\cA^{\ge 1}(C)$-module $\Hd_c(\QMN)$ as a
pushforward with support in $\bfL$.

In what follows, we will always include a cohomological shift by the
number $\delta=\frac12 \dim \Fix$ in the
definition of $\sigma_{*,\bfL}$. With this definition, a line bundle
on $\Fix$ pushes forward to object in zero cohomological degree.

§.§ Galois action

For applications to Eisenstein series, we will need cohomology of $\QMN$
with coefficients in local systems. Via the natural map

\begin{equation}
\label{eq:296}
\QMN(s: C \to \cP) \xrightarrow{\quad \NN_s \quad } \Pic(C)
\end{equation}

we can pull back a local system $\chi$ from $\Pic(C)$. We associate a
local systems on $\Pic(C)$ to characters $\chi$ of $\Pic(C)$ by requiring
that the geometric
Frobenius acts on the fiber over $\cO_C(x)\in \Pic_1(C)$ with eigenvalue

We have

\begin{equation}
\label{eq:297}
\NN_{s(x)} = \NN_s \otimes \cO_C(x)^2 \,.
\end{equation}

Therefore, we have a well-defined degree-preserving
Galois-equivariant correspondences

_c(C ×,    χ^2    ⊠χ)
_c(, χ)2  ,

_c(C ×,     _ℓ   ⊠χ)
_c(, χ)  ,

_c(C ×,    χ^-2    ⊠χ)
_c(, χ)-2  .

The cohomological shifts and Tate twists in these formulas appear
because pushforwards in the construction of $\be$ and
$\bff$ introduce a shift by $\sTate{2\delta}$, where
$\delta$ is the difference of
(virtual) dimensions between their target and domain.

To get rid of the cohomological shifts in the right-hand side of
(<ref>), we will center the cohomology of $\QMN$ around the
middle degree

\begin{equation}
\virdim \QMN = - 2 \deg \cL \,. \label{eq:308}
\end{equation}

For the eventual comparison with the spinor bundles, it is
a good idea to also symmetrize the module $\Hd_c(\QMN)$ with
respect to its grading by the degree $\deg \cL$ and also
with respect to the Galois action. Neither of these shifts
affects the $\fsl_2$-action.

We define

\begin{equation}
\label{eq:222}
\widehat{H}(\QMN,\chi) = \Hd_c(\QMN,\chi)\sTate{-2 \deg \cL+g-1} \otimes
\chi(\cK_C)^{-1}  \,.
\end{equation}

where the additional twist comes from

\begin{equation}
\label{eq:70}
\left(\Lambda^{\textup{top}} \Hd(C,\chi^2) \right)^{-1/2} =
\chi(\cK_C)^{-1} \sTate{g-1} \,.
\end{equation}

This is a character of the group $\Fr \times \bA\buu$ via the
action of Frobenius in the fiber of the local system $\chi$ on $\Pic(C)$ over
the canonical bundle $\cK_C$. As a function of $a$, it is a multiple of $a^{2g-2}$ in
accordance with (<ref>). See
(<ref>) below for more on the equality in (<ref>).

Note that the implicit dependence of $\chi(\cM)$ on $a \in \bA\buu$ will
also have a centering effect for the grading by $\bA\buu$, equivalently,
the grading by $\deg \cL$, which one can see in Lemma <ref>

Consider the Galois representation

\begin{equation}
\label{eq:299}
\begin{pmatrix}
\chi \\
& \chi^{-1}
\end{pmatrix} \in SL(2) \,,
\end{equation}

and the corresponding local system $\fsl_{2}^\chi$ obtained by
composing (<ref>) with the adjoint representation.
Künneth decomposition in (<ref>) results in the following

We have a Lie superalgebra action

\begin{equation}
\label{eq:43}
\Hd(C, \fsl_2^\chi) \otimes \widehat{H}(\QMN) \to
\widehat{H}(\QMN)\,,
\end{equation}

which is Frobenius-equivariant and degree-preserving.

Note the small, but important difference between (<ref>) and
(<ref>). With the symmetric normalization
(<ref>), the shift by $\rho$ as in  $\chi_\rho =
\chi\Tate{-\tfrac12}$ is not required. Of course, the shift will reappear in
any computation with actual Eisenstein series, because point counts
are given by Frobenius eigenvalues in $\Hd_c$, not in $\widehat{H}$.
One can also say that centering as in (<ref>) provides a more geometric way
to introduce the shift by $\rho$.

More generally, for moduli spaces of maps $f: C\to \bY$ to any
target $\bY$, we have

\[
\virdim = \deg f^*(T \bY) + \const = \left\langle c_1(\bY), \deg f
\right \rangle + \const\,,
\]

with the same constant $(1-g)\dim \bY$ in both instances. Thus
the coefficient $2$ in (<ref>) should, in general, be replaced by

We denote by $\cA_\chi(C)$ the algebra generated by $\Hd(C, \fsl_2^\chi)$.
As before, we have subalgebras

\begin{equation}\label{eq:38*3}
\cA_\chi(C) \supset \cA^{\ge 1}_\chi(C) \supset \cA^{2}_\chi
\end{equation}

generated by elements of cohomological degree $\ge 1$ and $2$,
respectively. Proposition <ref> generalizes as follows

We have a Frobenius-equivariant, degree-preserving

\begin{equation}
\label{eq:44}
\cA^{\ge 1}_\chi(C)\cong \Cliff_{\bX^\vee_0,\Gamma} \,.
\end{equation}

When $\chi_0^2 \ne 1$, there is a dramatic simplification in the
structure of these algebras. Indeed, the raising operators are
limited to

\begin{equation}
\label{eq:303}
\be\of{\gamma} \,  = \be(\gamma \boxtimes \, \cdot \,) \,,
\quad \gamma \in H^1(C,\chi^2_\rho) \,,
\end{equation}

and similarly for the lowering operators
$\bff\of{\gamma}$. In principle, the commutators of these operators
may involve $\bh\of{\bpt}$, which satisfies $\bh\of{\bpt}^2=0$ as
a corollary of (<ref>). We will see that, in fact,
$\bh\of{\bpt}$ acts by $0$ in (<ref>), except for one
exceptional case.

The vanishing $\bh\of{\bpt}$ of admits the following geometric
interpretation and generalization. If $\chi^2_0\ne 0$ then scheme-theoretic fixed locus
$\Fix_0 = \left(\bX^\vee_0\right)^{\Gamma'}$ is a length two scheme with
coordinate ring generated by $\bh\of{\bpt}$ modulo $\bh\of{\bpt}^2$.
The $\Gamma'$-fixed locus $\Fix = \left(\bX^\vee\right)^{\Gamma'}$ is,
however, reduced. Therefore, $\bh\of{\bpt}$ acts by zero in any
module in the image of $\sigma_*$, where

\begin{equation}
\sigma^*: \Cliff_{\bX^\vee_0,\Gamma}  \to \Cliff_{\bX^\vee,\Gamma}  \label{eq:43*2}
\end{equation}

is the pullback map. More generally, we have

\begin{equation}
\label{eq:45}
\Big( \chi^2_0\ne 0 \Big) \quad \Rightarrow \quad
\be\bff, \bff\be
\in \Ker \sigma^*\,.
\end{equation}

In particular, $\bh\of{\bpt} \in \Ker \sigma^*$ as a
commutator of classes in $H^1(C, \fsl_2^\chi)$.

§.§ Categorification of the L-genus

The Clifford algebra $\Cliff_{\bX,\Gamma}$ has a natural spinor bundle
$\Spin$ that may be constructed as follows. There is a canonical

\[
0 \to
\Omega^1_\hor \bX^\vee \to \Omega^1 \bX^\vee \to
\Omega^1_\ver \bX^\vee \to 0
\]

\[
\Omega^1_\hor \bX^\vee \cong p^* \left(T^*\bP^1 \right) \subset \Omega^1 \bX^\vee
\]

is the Lagrangian subbundle
formed by $1$-forms that vanish on the fibers of the projection $p:
\bX^\vee \to \bP^1$. It gives rise to an isotropic subbundle

\begin{equation}
H\Omega_{\bX^\vee,\hor} = H^1\left(C, \Omega^1_\hor \bX^\vee \right)\sTate{2} \subset
H\Omega_{\bX^\vee}\,, \label{eq:49}
\end{equation}

and hence to the spinor bundle

\begin{equation}
\label{eq:48}
\Spin = \sqrt{\det} \otimes \Cliff_{\bX^\vee, \Gamma} \Big/
\left(H\Omega_{\bX^\vee,\hor}\right) \,,
\end{equation}

where we mod out by the left ideal generated by (<ref>) and

\begin{equation}
\label{eq:66}
\det = \Lambda^{\textup{top}} H\Omega_{\bX^\vee,\hor}
= \left( \Lambda^{\textup{top}} H\Omega_{\bX^\vee,\ver}\right)^{-1}\,.
%  \left(\Hd\Omega_{\bX^\vee,\ver}\sTate{2} \right) \,.
\end{equation}

The existence of an equivariant square root is clear both when $\chi_0^2=1$ and
when $\chi_0^2\ne 1$. In the former case, we have the cohomology with
coefficients in a trivial local system, while in the latter case
$\dim \Fix=0$. See Section <ref>
for further discussion.

Following (<ref>), we define

\begin{equation}
\label{eq:51}
\cOh_{\Fix,\vir} = \Spin \sTate{-\delta}\,,
\end{equation}

\begin{equation}
\label{eq:52}
\delta = \tfrac 12 \dim \Fix =
\begin{cases}
1\,, \quad &\chi_0^2 = 1 \,, \\
0 \,, \quad  &\chi_0^2 \ne 1\,.
\end{cases}
\end{equation}

By construction, we have

\[
\left[ \cOh_{\Fix,\vir}  \right] = \bL(\bP^1) =
\bL_{\tfrac12}(T^*\bP^1) \,,
\]

which may be interpreted either as equality of classes in equivariant
K-theory of $\bX^\vee$, or as equality of distributions after pushing
forward to point modulo group. Our goal in this section is to lift
the relation between L-genera and constant terms to a relation between
$\widehat{H}(\QMN)$ and (<ref>).

Constant term is a function of $\cM\in \Pic(C)$ and in order to
introduce an $\cM$-dependent twist of $\Spin$ we need a homomorphism

\begin{equation}
\Pic(C)(\Bbbk) \owns \cM \mapsto \cO(\cM) \in
\Pic_{\Fr \times \bA\buu}(\bX^\vee)\,, \label{eq:25-2}
\end{equation}

such that the equivariant weights $\Fr \times \bA\buu$-weights of the
restriction of $\cO(\cM)$ to the fixed points $\{0,\infty\}$ are
$\chi(\cM)^{-1}$ and $\chi(\cM)$, respectively.

In general, $\Gamma'$, or any other group, will act nontrivially on the
restriction of the natural line bundles to the $\Gamma'$-fixed
locus. In particular, consider the line bundle $\cO(1)$ with
its natural linearization. This means that the group $\bA\buu =
\{ \diag(a^{-1},a)\}$ acts with the weights $a^{\pm 1}$ over
over the
fixed points $\{0,\infty\}$,
respectively. It follows from (<ref>) that an element
$\gamma\in \Gamma$ will act by $\chi(\gamma)^{\mp1}$
on the same fibers. Unless $\chi_0 =1$, this does not factor through

Therefore, if we want $\Fr$ to act on $\cO(1)$, we need
to pick its lift to $\Gamma/[\Gamma,\Gamma]$. We get such a lift if we fix a
point $x\in C$ of degree $1$ or, more generally, a divisor $\cM$
of degree $1$. We denote $\cO(\cM)$ the bundle $\cO(1)$ with the
corresponding action of $\Fr$. This factors through $\Pic_1(C)(\Bbbk)$
because $\chi_0$ is a character of $\Pic_0(C)(\Bbbk)$. By linearity,
this assignment extends to (<ref>).

Our main result in this section is the following

For any $\cM$ we have

\begin{equation}
\widehat{H}(\QMN) = \sigma_{*,\bfL} \left(\cOh_{\Fix,\vir} \otimes \cO(\cM)
\right)
%\Tate{\tfrac{\deg \cM+\delta}{2}}
\,,\label{eq:324}
\end{equation}

as a $\cA^{\ge 1}_\chi(C)$-module, unless

\begin{equation}
\label{eq:63}
(g,\cM,\chi_0^2) = (2,\cO_C, \textup{nontrivial}) \,.
\end{equation}

The proof of this theorem will occupy Section <ref>. The
exceptional case is analyzed in section <ref>  .
Among classical results to which Theorem <ref> may be
directly comparable we mention Semiinf2, Semiinf1.

§ PROOF OF THEOREM 2

§.§ Attracting manifolds and fixed loci

Consider the action of the group $\Aut(\cM) \cong \Gm$
on the space $\QMN$. It is clear, for instance from the description in section
<ref>, that it has two
basins of attraction as $z \in \Gm$ tends to $0 \in \overline{\Gm}$.

\begin{equation}
\Attr(0) = \{ s_2 = 0 \} \,, \quad \Attr(\infty) = \{ s_2 \ne 0 \} \,.
\label{eq:239}
\end{equation}

We denote the corresponding fix loci by

\[
\QMN^{\Gm} = \QM(0) \sqcup \QM(\infty) \,.
\]

On these fixed loci, we have $\cV = \cM \oplus \cO$
and $s$ is a quasisection of $\bP(\cM)$ or $\bP(\cO)$ respectively.
These are both $\bP^0$-bundles over $C$.

We observe that both $\QM(0)$ and  $\QM(\infty)$ are isomorphic
to the symmetric product  $\Sd = \bigsqcup \bS^d C$ or, equivalently,
the Hilbert scheme of points of $C$. Indeed, the isomorphism

\begin{equation}
\Sd C \to  \QM(\bP(\cM)) = \{ \cL, s \in \bP(H^0(\cM\otimes \cL^{-1}))\}
\label{eq:240}
\end{equation}

takes a divisor $D\in \Sd C$ to the quasimap

\[
\cL = \cM(-D) \xrightarrow{\quad} \cM \,.
\]

The following table

\begin{equation}
\label{eq:305}
\begin{tabular}{ c| ccc}
& $\NN_s$ & $N_{\Attr/\textup{\textsf{fixed}}}$ & $\rk
\\ [0.5ex]
\hline
$\QM(0)\phantom{\Big|}$ & $\cM^{-1}(2D)$ & $-\Hd(\cM)$ & $-\deg \cM +
$\QM(\infty)$ & $\cM(2D)$ & $\Hd(\cM(D))-\Hd(\cM)$ & $\deg D$\\
\end{tabular}
\end{equation}

collects the information about the normal bundle (<ref>)
to the fixed sections and about the normal bundle to the fixed
locus inside the attracting manifold. In particular,

_c((∞))    = ((∞))       -2D  ,

_c((0))    = ((0))       2+2-2g

Consider the
long exact sequence of a pair

\begin{equation}
\label{eq:242}
\xymatrix{
& \Hd_c(\QMN) \ar[dr] \\
\Hd_c(\Attr(\infty))  \ar[ur] && \Hd_c(\Attr(0))
%\ar[ll]_{[1]} }
<EMAIL_ADDRESS>}
\end{equation}

for cohomology with compact supports. Since both
cohomology groups in (<ref>) are pure, we
conclude the following

The connecting dotted map in (<ref>) vanishes.

The fix loci have their own correspondences as in (<ref>),
which we denote by $\be^{(0)}$ and $\be^{(\infty)}$, respectively.
The main difference between quasisections of $\bP^k$-bundles
for different $k$ is  the rank of normal bundle $\cN$ as in
(<ref>). This rank equals $k+1$ and hence $c_{k+1}(\cN)$ is
in the definition of the correspondences. In particular, the
analog of (<ref>) for $\bP^0$-bundles reads

\begin{equation}
[\iota^* , \iota_{*} ] \circ \Delta_{C,*}  = c_1(\cN
\otimes T C) - c_1(\cN)
= \tau\,. \label{eq:224*2}
\end{equation}

\begin{equation}
\label{eq:241}
\left[ \be^{(0)}, \bff^{(0)} \right] = (\textstyle{\int_C}
\otimes 1) \,  \Delta_C^*  \,,
\end{equation}

meaning that these correspondences generate a Heisenberg–Clifford
algebra $\Heis$ in place of $\fsl_2$.

As before, a local system $\chi$ defines a local system $\Heis_{\chi}$, in which
$\be$ and $\bff$ are twisted by $\chi$ and $\chi^{-1}$,
respectively. Note that,  in accordance with the first column in
(<ref>) and the Tate shift that
is present in (<ref>), we need the local system
$\Heis_{\chi^2}$ on $\QM(0)$ and the local system $\Heis_{q\chi^2}$
on $\QM(\infty)$ to make the triangle
(<ref>) Galois-equivariant.

The following result is well-known and constitutes a $1$-dimensional analog
of Nakajima's action on the Hilbert schemes of points of surfaces

$\Hd(\Sd C, \chi)$ is a Verma module for $\Hd(C,\Heis_{\chi})$.

For the trivial local system, this goes at least as far back as the
work of Macdonald <cit.>. For a nontrivial local, this is a
consequence of representation theory of Clifford algebras and the

If $\chi$ is nontrivial then $\Hd(\bS^d C, \chi)=0$ for $d>2g-2$.

The map $S^d C \to \Pic_d(C)$ is projective bundle for $d > 2g-2$ and
the pushforward of the constant sheaf under this map forms a trivial
local system on the base. Since the local system $\chi$ is pulled
back from the base, and the cohomology of an abelian variety with
nontrivial coefficients vanishes, the claim follows.

As a corollary of the proof, we note that $\Hd(\bS^{2g-2} C, \chi)$
vanishes except for

\begin{equation}
\label{eq:67}
H^{2g-2}(\bS^{2g-2} C, \chi) = \chi(\cK_C) \otimes
H^{2g-2}(\bP^{g-1}) \,,
\end{equation}

where $\cK_C$ is the canonical bundle of $C$ and
$\bP^{g-1} \subset \bS^{2g-2} C$ is the canonical linear system. From
this and Proposition <ref> one may observe the
classical fact that

\begin{equation}
\Lambda^{\textup{top}} \Hd(C,\chi) =
\chi(\cK_C)\sTate{2-2g}\label{eq:68}
\end{equation}

for any character $\chi$, trivial or not.

§.§ Localization is localization

Equivariant localization in topology
relates  equivariant cohomology of a space, ordinary and
extraordinary, with the cohomology of the fixed loci.
In our case, this comparison is provided by exact
sequence (<ref>) and the isomorphisms

Using these isomorphisms, we may express the
$\fsl_2$-action in terms of the Heisenberg operators.
Our goal in this subsection is to identify this topological
equivariant localization with localization in
representation theory as discussed in section

The map $\Hd_c(\QMN) \to \Hd_c(\Attr(0))$ is pullback under
the embedding and thus commutes with the pullback $\iota^*$.
For the pushforward, we need to look at the normal bundle $\cN$,
which fits into the sequence

\begin{equation}
\label{eq:326}
0 \to \underbrace{\cM \otimes \cL^{-1} \otimes
\cO_{\QM}(1)}_{\cN^{(0)}}\to  \cN \to  \underbrace{ \cL^{-1} \otimes
\cO_{\QM}(1) }_{\cN^{(\infty)}}\to 0 \,.
\end{equation}

The map $\iota^{(0)}_*$ lacks the quotient term in (<ref>),
the following

The diagrams

\begin{equation}
\label{eq:243}
\xymatrix{
\Hd_c(C\times \QMN) \ar[r]  &\Hd_c(C\times \Attr(0)) \\
\Hd_c(C\times \QMN) \ar[r]  \ar[u]^{\iota^*} & \Hd_c(C\times \Attr(0))
\ar[u]_{\iota^{(0),*}}
} \qquad
\xymatrix{
\Hd_c(C\times \QMN) \ar[r] \ar[d]_{\iota_*}  &\Hd_c(C\times \Attr(0))
\ar[d]^{\square_*}\\
\Hd_c(C\times \QMN) \ar[r]  & \Hd_c(C\times \Attr(0))
\end{equation}

\[
\square_* = c_1(\cN^{(\infty)}) \, \iota^{(0)}_*= \iota^{(0)}_*
c_1(\cN^{(\infty)}\otimes TC)\,,
\]

Note these formulas are consistent with interpreting the
virtual bundle

\[
N_{\QMN/\Attr(0)} = \Hd(C,\cL^{-1}) \otimes \cO_{\QM}(1) \,.
\]

as the virtual normal bundle to $\Attr(0)$ in $\QMN$.  The
correspondence $\iota^{(0)}$ changes it by

\[
\iota^{(0),*} N_{\QMN/\Attr(0)}  - N_{\QMN/\Attr(0)} =
\cN^{(\infty)}\otimes TC \,.
\]

The normal bundle

\begin{equation}
\label{eq:245}
N_{\QMN/\QM(\infty)} = \left( \Hd(\cM\otimes \cL^{-1}) -
\Hd(\cM)\right) \otimes \cO_{\QM}(1)
\end{equation}

is a vector bundle of  rank $-\deg \cL$ over
The map $ \Hd_c(\Attr(\infty)) \to \Hd_c(\QMN)$ commutes
with $\iota_*$, since both maps are pushforwards. For pullbacks, we
need to look at the effect of $\iota^{\infty}$ on the normal bundle
(<ref>). We deduce

The diagrams

\begin{equation}
\label{eq:246}
\xymatrix{
\Hd_c(C\times\Attr(\infty))  \ar[r] & \Hd_c(C\times\QMN)  \\
\Hd_c(C\times\Attr(\infty))
\ar[u]^{\square^*}  \ar[r]  & \Hd_c(C\times\QMN) \ar[u]^{\iota^*}
} \quad
\xymatrix{
\Hd_c(C\times\Attr(\infty))
\ar[d]_{\iota^{(\infty)}_*} \ar[r] &  \Hd_c(C\times\QMN) \ar[d]^{\iota_*}  \\
\Hd_c(C\times\Attr(\infty)) \ar[r]   & \Hd_c(C\times\QMN)
\end{equation}

\[
\square^* = c_1(\cN^{(0)}\otimes TC) \, \iota^{(\infty),*} =\iota^{(\infty),*}
\]

Note that the nontrivial squares in (<ref>) and (<ref>)
also follow from the trivial ones and the relations (<ref>)

One has the following analog of the Casimir element equation

\begin{equation}
\label{eq:55}
\bff^{(0)}_{13} \, \be^{(0)}_{23} \circ \Delta_{C,*} = \bh^{(0)}
\eqdef  (\textstyle{\int_C} \otimes 1) \circ c_1(\cN^{(0)} \otimes
TC ) \,.
\end{equation}

Indeed, (<ref>) is just the composition of pushforward and
pullback by the same map. We can use (<ref>) to  expand out (<ref>) along the
lines of (<ref>).

We have

\begin{align}
\be^{(0)}\of{\bpt} \, \bff^{(0)}\of{\bpt} &=\eta \,, \notag\\
\be^{(0)}\of{\bpt} \, \bff^{(0)}\of{\alpha} + \be^{(0)}\of{\alpha} \, \bff^{(0)}\of{\bpt}  &=\alpha
\label{eq:306-0}
\end{align}

where $\alpha \in H^1(C)=H^1(\Pic(C))$ acts on $\QMN$ by

For brevity, let us denote by $FE$ the following composition

\begin{equation}
\label{eq:53}
FE =  \bff_{13} \, \be_{23}  \circ \Delta_{C,*} = c_2(\cN \otimes TC)
\end{equation}

of pushforward $\be$ and the pullback $\bff$. Let
$FE^{(0)}$ denote the analogous expression appearing in (<ref>). From (<ref>),
we conclude

FE     = FE^(0)
c_1(^(∞)⊗TC)       =
c_1(^(∞)⊗TC)   FE^(0)

= FE^(∞)   c_1(^(0)⊗TC)
=  c_1(^(0)⊗TC)   FE^(∞)    ,

which provides a different computations of the nontrivial squares in
(<ref>) and (<ref>).

Writing out the equations (<ref>) using
(<ref>), we obtain

\begin{align}
\bff\of{\bpt} &= \bff^{(0)}\of{\bpt} \notag = Y \\
\be\of{\bpt} &= \eta \, \be^{(0)}\of{\bpt} = X^2 Y \notag \\
\bff\of{\alpha} &= \bff^{(0)}\of{\alpha} = dY \notag \\
\be\of{\alpha } &= \eta \, \be^{(0)}\of{\alpha} + \alpha \,
\be^{(0)}\of{\bpt} = 2 XY dX + X^2 dY \,,
\label{eq:57}
\end{align}

where we use the shorthand

\[
X = \be^{(0)}\of{\bpt}\,, \quad Y=\bff^{(0)}\of{\bpt}\,, \quad
dX = \be^{(0)}\of{\alpha}\,, \quad dY=\bff^{(0)}\of{\alpha} \,.
\]

These formulas have a geometric interpretation which may be
illustrated using Figure (<ref>). Let $X$ and $Y$ be the
coordinates around $0 \in \bX^\vee$ with weights $a^{-2}$ and $q a^2$,
respectively. Clearly, the first two lines in (<ref>) describe
the map $\sigma$ in the coordinates $X$ and $Y$, while the second two
lines describe the action of $\sigma^*$ on $\Omega^1$.

\[
\textup{Chart}_0= \bX^{\vee}\setminus T^*_\infty \,, \quad
\textup{Chart}_\infty = \bX^{\vee}\setminus T^*_0 \,,
\]

be two affine charts covering $\bX^\vee$, where $T^*_0$ and $T^*_\infty$ denote the cotangent fibers over $0,
\infty \in \bP^1$. We rephrase (<ref>) as follows

The equivariant localization homomorphism

\[
\cA^{\ge 1}(C) \to \cA^{(0),\ge 1}(C) \oplus \cA^{(\infty),\ge
\]

given in (<ref>) and (<ref>) coincide with the
natural maps of the Clifford algebras

\[
\Cliff_{\bX^\vee_0} \xrightarrow{\quad \textup{restriction} \, \circ \,
\sigma^*\quad }  \Cliff_{\textup{Chart}_0} \oplus
\Cliff_{\textup{Chart}_\infty}\,.
\]

This forms a direct link between equivariant localization and
localization in representation theory.

§.§ The stable range

Our next goal is to prove Theorem <ref> outside the vertex of the
cone, that is, one the set where $\sigma$ is an isomorphism.
This set is covered by two affine charts

\[
\bX_0^\vee \setminus \{0\} = \{\textup{$\be\of\bpt$ is invertible} \}
\cup
\{\textup{$\bff\of\bpt$ is invertible} \} \,.
\]

Since the degree of a quasimap is always bounded below, the lowering
operators $\bff$ are locally nilpotent. Thus both sides of equality
in Theorem <ref> restrict to zero on the chart where
$\bff\of\bpt$ is invertible.

Since $\be\of\bpt$ is a raising operator, inverting it involves
understanding of the $\fsl_2$-action in the stable range $\deg \cL
\ll 0$. We will see
in Section <ref> that
all cohomology vanish in this range if $\chi_0^2\ne 1$. Therefore, our
focus in this section is on cohomology with trivial local system

If $\be\of\bpt$ is invertible then the Casimir relation (<ref>)
determine the action of $\bff$ from the action of $\be$ and $\bh$,
that is, from the action of the Borel subalgebra $\mathfrak{b} \subset
\fsl_2$.
We recall from Proposition <ref> that the operators $\bh\of\bpt$
and $\bh\of\alpha$ act by a multiple of $\eta$ and $\alpha$,
respectively, where $\alpha\in H^1(C) = H^1(\Pic C)$ acts by cup
product with the pullback of $\alpha$ to $\QMN$. The commutator

\begin{equation}
\label{eq:66*2}
\left[ \alpha_1, \be\of{\alpha_2} \right] = \be\of{\alpha_1 \cup \alpha_2}
\end{equation}

is the only nontrivial commutator in $\mathfrak{b}\otimes H^{\ge

Let $\cF$ be any locally free Clifford module on $\bX_0^\vee$. It is
clear from the discussion in Section <ref> that

\begin{equation}
H^1_{\bfL}(\bX_0^\vee \setminus \{0\},\cF)
\cong \left(\Q_\ell[\eta^{\pm 1}] \big/ \Q_\ell[\eta] \right) \otimes
\underbrace{\cF \Big|_{\bfL_\infty \setminus \{0\}}}_{\Ker
\eta} \,, \label{eq:67*2}
\end{equation}

as $\cA^{\ge 1}$-modules, while $H^i_{\bfL} =0$ for $i\ne 1$. One may
call vectors annihilated
by $\eta$ primitive. We say that a module of the form (<ref>) is
freely $\eta$-cogenerated by its primitive elements.

The $\cA^{\ge 1}$-module $\widehat{H}(\QMN)$ with $\be\of\bpt$ inverted
is freely $\eta$-cogenerated by its primitive elements. The primitive
elements are freely generated under the action of the operators
$\be\of{\bpt}^{\pm 1}$ and $\be\of\alpha$,
$\alpha \in H^{1}(C)$,
by any $1$-dimensional subspace of the form
$H^{\textup{top}}_c(\QMN_d)$, where $d=\deg \cL \ll 0$.

Evidently, $H^{\textup{top}}_c(\QMN_d)$ is annihilated by cup products
by $\alpha \in H^1(C)$. Thus, the above proposition and (<ref>)
completely determines the structure of this module. The shifts and
twists in definition of both sides of (<ref>) are
arranged so that primitive elements equal the
restriction of $\Spin \otimes \cO(\cM) \Tate{\tfrac{\deg
\cM+\delta}{2}}$ to $\bfL_\infty \setminus \{0\}$. Therefore,
we have the following

Theorem <ref> is true outside the vertex of the

The proof of Proposition <ref> will occupy the rest of this

Let $\QM_d \subset \QM$ denote the component with $\deg \cL = d$,
and similarly for $\QMN_d$. We recall from Section <ref>
that $\QM_d \to \Pic_d(C)$ is a projective bundle for $d\ll 0$. Similarly, $\QMN_d$ is
the quotient of a projective bundle over $\Pic_d(C) \times
H^1(\cM)$ by the action of the unipotent group $H^0(\cM)$.
By the projective bundle theorem, we have

\begin{equation}
\cong
\underbrace{\Ld \, H^1(C)}_{\textup{coefficients}} \, \big[\eta\big]
\Big/ \bC_{\cO\oplus \cM} (\eta)
\,, \quad d \ll 0 \,, \label{eq:54}
\end{equation}

as a module over $\eta$ and $H^1(C)$, where

\[
\bC_{\cV} (\eta) = \textup{Chern
polynomial}(H^0(C,\cV\otimes\cL^{-1}),\eta) \,.
\]

While we can argue with just abstract properties of this Chern
polynomial, it is easy to describe it concretely.

Since $H^0(C,\cV\otimes \cL^{-1})$ is obtained by pushforward
along $C$, which is $1$-dimensional, we conclude that

\[
\textup{ch}_i \left(H^0(\cV\otimes \cL^{-1})\right) =0 \,, \quad i>1 \,.
\]

On the other hand, by the definition and symmetry of the theta divisor
$\Theta$, we have

\begin{equation}
\label{eq:59}
c_1 \left(H^0(\cV\otimes\cL^{-1})\right) = \rk(\cV) \, \Theta\,,
\quad \Theta = \sum_{i=1}^g \gamma_i
\gamma^\vee_i\,.
\end{equation}

\begin{equation}
\label{eq:61}
\bC_{\cV} (\eta)= \eta^{\rk H^0(\cV \otimes \cL^{-1})}
e^{-\textup{ch}_1/\eta} = \eta^{-rd + \const}
\prod_{i=1}^g (\eta - r \gamma_i
\gamma^\vee_i) \,, \quad r = \rk(\cV) \,,
\end{equation}

where the constant equals $\deg \cV + r-(r+1)g$. Importantly, this
stabilizes in that it depends on $d$ only through the $\eta^{-rd}$

It is easy to describe the action of the operators $\be$ and $\bff$ in
the stable range. Let $P=P(\eta, \alpha_1,\alpha_2, \dots)$ be an
element of the ring in the right-hand side of (<ref>).

We have

P    = -P  ,        α  P    = -
∂/∂α^∨ P  ,

P    = η^2 P  ,        α  P    =
η^2 ∂/∂α^∨ P - 2 ηαP .

The pullback $\iota^*$ is an algebra homomorphism which acts by

ι^*(1⊗α)    = α⊗1 + 1 ⊗α ,
α∈H^1(C)  ,

ι^*(1⊗η)     = ⊗1 + 1 ⊗η ,

where $\bpt\in H^2(C)$ is the generator. The relation (<ref>)
was already discussed in the proof of Proposition <ref> . To see (<ref>),
note that $\eta$ is represented by the divisor where a linear function of
$s(x)$ vanishes for some fixed $x\in C$.

The equations (<ref>) follows from the definition of $\bff$ and
the above description of $\iota^*$. Formulas (<ref>)
follow from the Casimir relations (<ref>).

The line $H^{\textup{top}}_c(\QMN_d)$ is spanned by $\eta^{\max}
\Lambda^{\textup{top}} H^1(C)$, where the maximal power of $\eta$ is
one less than the degree of $\bC(\eta)$. It is clear that the action
of $\be\of\bpt^{\pm1}$ takes all these elements to each other.

looking at the lowest degree term in the odd variables, it is clear
that the action $\be\of\alpha$ generates a $2^{2g}$-dimensional
space of primitive vectors from $H^{\textup{top}}_c(\QMN_d)$.
These have to exhaust all primitive vectors and this completes
the proof of Proposition <ref>.

In fact, it is not difficult
to see directly that the product of all $\be\of\alpha$, applied to $H^{\textup{top}}_c$ reproduces
precisely the relation $\bC(\eta)$.

§.§ Completion of the proof

By Proposition <ref>, the exact sequence  (<ref>)
translates into the exact sequence

\begin{equation}
\label{eq:310}
0 \to \textup{Verma}\to
\widehat{H}(\QMN) \to \textup{dual Verma} \to
0 \,,
\end{equation}

where we call a $\cA_\chi(C)$-module a Verma module if it is free as a module over
the commuting operators $\be$, while a dual Verma
module is dual to a free module as a module over the commuting
operators $\bff$. Note this definition also involves operators coming
from $H^0(C)$.

Our plan is to compare this sequence with the exact sequence
of the pair

\begin{equation}
\label{eq:71}
0 \to \sigma_{\bfL_\infty,*} \to \sigma_{\bfL,*} \to
\sigma_{\bfL_0,*} \to 0 \,,
\end{equation}

as in (<ref>). We will first compare the submodules and
quotient modules, and then compare the extensions.

It is clear from Proposition <ref> that the terms of the
filtration (<ref>) for $\sigma_{\bfL,*} \cOh_{\Fix,\vir} \otimes \cO(M)$
are similarly
a Verma
module and a dual Verma module. Therefore, it suffices to compare
the weights of the generators. This is done in the following

For both $\widehat{H}(\QMN)$ and $\sigma_{\bfL,*} \cOh_{\Fix,\vir} \otimes
\cO(M)$, we have

= χ(⊗_C^-1)
g-1  ,

= χ(^-1⊗_C^-1)
1-g    .

For $\widehat{H}(\QMN)$, this is an immediate consequence of
the definition (<ref>) and table (<ref>). In the
computations with the spinor bundle, we will need the number
$\delta= \delta_{\{\chi_0^2=1\}} \in \{0,1\}$, where $
\delta_{\{\chi_0^2=1\}}$ denotes the indicator function. From
we deduce

\begin{equation}
\label{eq:60}
\left(\Lambda^{\textup{top}} H^1(C, \chi^2)\right)^{1/2} = \chi(\cK_C)
\chi^{2\delta} \sTate{1-g-\delta} \,.
\end{equation}

Consider the bundle $\Spin$. The generator of the submodule
and the quotient modules come from the elements $1\in \Cliff$ and
the product of all $\bff\of{\alpha_i}$,
respectively. With the square root twists, these have the weight
$ \left(\Lambda^{\textup{top}} H^1(C, \chi^2)\right)^{-1/2}$
and $\left(\Lambda^{\textup{top}} H^1(C, \chi^{-2})\right)^{1/2}$,
respectively. These weights form the first row in the following

\begin{equation}
\label{eq:305-2}
\begin{tabular}{ c| cc}
\textup{contributions from}  & \textup{quotient} & \textup{sub}
\\ [0.5ex]
\hline
$\Spin $  & $\chi(\cK_C)^{-1}
\chi^{-2\delta} \sTate{1-g-\delta}$& $\chi(\cK_C)^{-1}
\chi^{-2\delta} \sTate{\delta+g-1}$\\
$\cOh_{\Fix,\vir} $  & $\sTate{-\delta}$& $\sTate{-\delta}$\\
$\cO(\cM)$  & $\chi(\cM)^{-1}$& $\chi(\cM)$\\
\textup{local cohomology}  & $\chi^{2\delta}\sTate{2\delta}$& $\chi^{2\delta}$\\
\end{tabular}
\end{equation}

where the shifts in local cohomology appear as in the computation done
in section <ref>. Collecting terms in the above table
the proof.

To finish the proof, we need to identify the classes of the
extension. If $\chi^2_0 = 1$, then the submodule is free as
a module over $\be\of\bpt$. This means the extension class
is uniquely determined by its localization with respect to
$\be\of\bpt$. By Proposition <ref>, the extension
classes coinside after such localization, completing the
proof in the $\chi^2_0 = 1$ case.

Consider the case $\chi^2_0 \ne 1$, in which case the
statement of the theorem is equivalent to the splitting of the
module $\widehat{H}(\QMN, \chi)$. The submodule
is a free module over $\be\of{\alpha_i}$, with the operators
$\bff\of{\alpha_j}$ acting by zero. For the quotient module,
the situation is reversed. Thus $\bff\of{\alpha_j} \, \be\of{\alpha_i} =
0$ for any $\alpha_i$ and $\alpha_j$, and the module splits
if and only if the product in the other order also vanishes. This is
equivalent to the vanishing of the commutator $\bh\of\bpt$.

Both the sub and the quotient module are nonzero for a single
cohomological degree for given degree of the quasimap.
From the considerations of degrees, we see that
$\rk \bh\of\bpt\le 1$. Since $\bh\of\bpt$ is central, we conclude

\[
\Image \bff\of{\alpha_i} \subset \Ker \bh \of \bpt \,,
\quad \Image \bh \of \bpt \subset \Ker \be\of{\alpha_i} \,,
\]

for any $\alpha_i$.
Therefore, if nonzero, $\bh\of\bpt$ has to map the generator of the
quotient to the cogenerator of the submodule. Comparing the
degrees, we see that this is only possible if

\[
g=2 \,, \quad \deg \cM = 0 \,.
\]

This case will be analyzed explicitly below and, indeed,
$\bh\of\bpt\ne 0$ when $\cM=\cO_C$.

§.§ The exceptional case

Consider the case $g=2$, $\cM=\cO_C$. In this case, the generator of
the quotient module and the cogenerator of the submodule both
correspond to the class of $H^2(\bS^2 C,\chi)$ represented by
the canonical linear system $\bP^1 \subset \bS^2 C$. This cycle
lies in the fiber

\[
\xymatrix{
\QMN_{\cK^{-1}} \ar@{^{(}->}[r] \ar[d] & \QMN_{-2} \ar[d] \\
\cK^{-1}_C \ar@{|->}[r] & \Pic_{-2}(C) \,,
\]

over the point $\cL = \cK^{-1}_C $. The moduli space  $\QMN_{\cK^{-1}}$ is formed by all possible extensions

\[
0 \to \cO \to \cV \to \cO \to 0
\]

together with a section

\[
s \in H^0(\cK \otimes \cV)
\]

taken modulo the action of $\Gm \times H^0(\cO) \subset \Aut(\cV)$.
The operator $\bh\of\bpt$ acts by a multiple of $\eta = c_1(\cO(1))$.
It descends from the same action on the space $\widetilde{\QMN}_{\cK^{-1}}$, where we divide
by the action of $\Gm$ only.

Consider the exact sequence

\[
0 \to H^0(\cK)  \to H^0(\cK \otimes \cV) \to  H^0(\cK)
\xrightarrow{\quad \beta_\cV \quad} H^1(\cK)
\]

where the map $\beta$ is the pairing with the class of
the extension

\[
\beta_\cV \in \Ext^1(\cO,\cO) = H^0(\cK)^*  \,.
\]

Possible bundles $\cV$ are precisely parameterized by
their classes $\beta_\cV$. Denoting $K=H^0(\cK)$, we obtain
the following description of $\widetilde{\QMN}_{\cK^{-1}}$
as a subset of $\bP(K^{\oplus
2}) \times K^*$

\begin{equation}
\xymatrix{
\bP(K\oplus \beta^\perp) \ar@{^{(}->}[r] \ar[d]
&\widetilde{\QMN}_{\cK^{-1}} \ar[d] \\
\beta \ar@{|->}[r] & K^* \,. } \label{eq:62}
\end{equation}

The action of $\Gm = \Aut \cM$ corresponds to scaling the first copy
of $K$ in $K^{\oplus 2}$.

In terms of the diagram (<ref>), we generator of the quotient
module and the cogenerator of the submodule are dual to
the fundamental cycle in $H_8$ and
the cycle $\left[\bP(K)\right]\in H_6$, respectively. Similarly,
$\eta$ is dual to $\left[\bP(\textup{line} \oplus \beta^\perp) \right] \in
H_6$. One can move the line off the origin and then send it to
infinity by the action of $\Gm = \Aut \cM$. Thus

\[
\left[\bP(K)\right] = \left[\bP(\textup{line} \oplus \beta^\perp)
\right]
\]

and so $\bh\of\bpt\ne 0$ in this instance.

If $\cM\not\cong \cO$ then the classes corresponding to $H^2(\bS^2 C,\chi)$
lie over different fibers of the projection to $\Pic_{-2}(C)$, thus
cannot possible be related via multiplication by $\eta$. This
the proof of Theorem <ref>.

§ APPENDIX. RANK 2 BUNDLES OVER $\BP^1$

§.§ $\Bun_\bG$ and $\Gr_\bG$

This appendix is elementary and written for readers with no background in
automorphic forms. Our main objective
here is to explain by means of the simplest example how spectral
decompositions are encoded by certain additive decompositions of
distributions. We also explain what these distributions have to do
with constant terms of Eisenstein series. Finally, we point out
how this elementary example fits into the general story told in our

Let $C=\bP^1$ be the projective line over a finite field $\Bbbk$ with
$q$ elements.
More generally, one may replace $\bP^1$ by a smooth
projective curve over
$\Bbbk$ which remains irreducible over the algebraic
closure $\overline{\Bbbk}$, and that would be setting of
Section <ref>. We denote by $\F =
\Bbbk(C)$ the field of rational functions on $C$. For
$C=\bP^1$, this simply means $\F= \Bbbk(t)$, where
$t$ is an indeterminate.

A closed point $x\in \bP^1$ other than $\infty \in \bP^1$ corresponds
to an irreducible monic polynomials in $t$. The degree $\deg_x$ is
the degree of this polynomial. We denote by $\Bbbk_x$ the
corresponding finite extension of $\Bbbk$. Consider
the completion $\F_x$ of $\F$ at $x$ and its subring

\begin{equation}
\label{eq:172}
\Bbbk_x(\!(t)\!) \cong  \F_x \supset  \bO_x  \cong \Bbbk_x[\![t]\!]
\end{equation}

by functions regular at $x$. Here $\Bbbk_x(\!(t)\!)$ and $\Bbbk_x[\![t]\!]$
denote formal Laurent and Taylor series, respectively.
One defines integral adeles and all adeles of $\F$,
respectively, by

\begin{equation}
\bO = \prod_{x\in C} \bO_x \subset  {\prod_{x\in C}}' \, \F_x = \A \label{eq:152}
\end{equation}

where the prime in (<ref>) means that any adele is integral for
all but finitely many $x$.

A vector bundle $\cV$ over $C$ may be trivialized in the formal
neighborhood of each closed point $x$ and also at the generic point.
The corresponding transition function identifies isomorphism classes
of vector bundles with
double cosets

\begin{equation}
\Bun_\bG = \bG(\F) \backslash \bG(\A) /\bG(\bO) \,, \label{eq:2}
\end{equation}

where $\bG=GL(n)$, $n= \rk \cV$.

For any reductive $\bG$, the
double cosets (<ref>) describe the $\Bbbk$-points of the stack $\cBun_\bG$ of
$\bG$-bundles on $C$. For

\[
\bG=  SL(n), PGL(n)
\]

this is the same as vector bundles with trivial
determinant, and vector bundles up to tensoring with a line bundle,
respectively. The latter can be also identified with
$\bP^{n-1}$-bundles $\cP$ over the curve $C$.

A very special property of $C=\bP^1$ is that every vector bundle is a
direct sum of line bundles. As a result, the
structure of $\Bun_{\bG}$, and the corresponding theory of automorphic
forms, becomes almost trivial. Concretely,

\begin{equation}
\Bun_{SL(2)} = \left\{ \cO(m) \oplus \cO(-m) \right\}_{m\ge 0}
\hookrightarrow \Bun_{PGL(2)} = \left\{ \cP_k
\right\}_{k\ge 0} \,, \quad \cP_k = \bP\left(\cO(k) \oplus \cO\right)
\,, \label{eq:153}
\end{equation}

where the map takes a rank 2 bundle to associated
$\bP^1$-bundle and its image consist of $\cP_k$ with $k$ even.

For $C=\bP^1$, there are no cusp forms, no Galois
representations, Eisenstein series are equal to their constant term,
etc. Luckily, the structures that we want to illustrate survive even
in this simplest case and can give an idea about how things work in

Let $x \in C$ be a point of degree 1.
The bundles in (<ref>) are trivial in the complement of $x$,
which means that

\begin{align}
\Bun_\bG &= \bG(\cO_{C \setminus x}) \backslash \bG(\F_x)
/\bG(\bO_x) \notag \\
&= \bG(\Bbbk[t^{-1}]) \backslash \bG(\fst)
/\bG(\fTst) \label{eq:154}\,,\\
& = \bG(\Bbbk[t^{-1}]) \backslash \Gr_\bG \notag
\end{align}

\begin{align}
\label{eq:155}
\Gr_\bG = \bG(\fst)
/\bG(\fTst) \,,
\end{align}

are the $\Bbbk$-points of the affine Grassmannian.
Among other
things, $\Gr_\bG$ describes $\bG$-bundles over $\bP^1$ trivialized
in the complement of a $\Bbbk$-point $x$.

For $\bG = GL(n)$, we have

\[
\Gr_{GL(n)} = \{ \textup{$\fTst$-lattices $L\subset
\fst^{\oplus n}$}\}\,.
\]

The lattice $L(\cV)$ corresponding to a  vector bundle $\cV$ is
described as follows

\begin{equation}
\label{eq:157}
\xymatrix{
H^0(\bD_x,\cV) \, \ar@{^{(}->}[r] \ar@{=}[d]&
H^0(\bD^\times_x,\cV) \ar@{=}[d] \\
L(\cV) \, \ar@{^{(}->}[r] & \fst^{\oplus n}
} \,,
\end{equation}

\[
\bD_x = \Spec \Bbbk[\![t]\!]\,, \quad
\bD^\times_x = \Spec  \Bbbk(\!(t)\!) \,,
\]

are the formal neighborhood of $x$ and
the punctured formal neighborhood of $x$.
For $\bG = SL(n)$ and $\bG =  PGL(n)$ one considers lattices $L$
such that $\Lambda^n L = \fTst$ and lattices up to scale,

For $\bG=PGL(2)$, the points of (<ref>) correspond to vertices
of a regular $(q+1)$-valent tree, see <cit.>. This is the rank 1 case
of the general Bruhat-Tits building. Edges $L\textup{---}L'$ of this tree correspond to
pairs of lattices such that

\[
L \supsetneq L' \supsetneq t L \,.
\]

Note this is a symmetric relation.
For $q=2$, the tree is plotted in

\begin{gather}
\label{eq:156}
\raisebox{-100pt}{\includegraphics[height=200pt]{Tree1.pdf}} \\
\textup{\sl Orbits of $\bG(\fTst)$ on $\Gr_\bG$ for $q=2$} \notag
\end{gather}

In (<ref>), vertices of the tree are represented by circles and their color
reflects the distance from the root vertex ($=$ the trivial bundle).

The stabilizer of the root vertex in
$\bG(\fst)$ is $\bG(\fTst)$,
which acts transitively on the sphere of each radius around the
root. In other words, any path without backtracking
from the root to infinity gives a fundamental domain for $\bG(\fTst)$.
Three such paths are plotted in (<ref>). The tree in
(<ref>) is the nonarchimedian analog of the unit disk,
upper half-plane, and the action of $\bG(\fTst)$ should be compared
to the action of the maximal compact subgroup $SO(2,\R) \subset

The subset

\[
\Gr_{SL(2)} \subset \Gr_{PGL(2)}
\]

is formed by vertices at even distance from the root.

§.§ Hecke operators

The spherical Hecke algebra

\begin{align}
\SH_x = \,\, & \textup{compactly supported functions} \notag \\
&\textup{on $\bG(\fTst) \backslash \bG(\fst)/  \label{eq:159}
\bG(\fTst)$}
\end{align}

acts by right convolution on function on $\Gr_\bG$. It is
clear from the description of orbits in (<ref>) that $\SH_x$
for $\bG = PGL(2)$
is an algebra of polynomials in one generator $\Delta$ which may be
chosen as the  graph Laplacian

\begin{equation}
\left[\Delta f \right] (L) = \sum_{L\textup{---}L'}
f(L') \label{eq:158} \,.
\end{equation}

The spherical Hecke algebra for $\bG=SL(2)$ is spanned by even powers of

In terms of $\bP^1$-bundles $\cP \to C$, the action of $\Delta$ is
given by modifications as in (<ref>) in Section <ref>.

A section $s$ of $\cP$ in
determines a unique section $s'$ of $\cP'$. Self-intersections of
$s$ and $s'$ are related by

\begin{equation}
\label{eq:161-2}
(s',s') - (s,s) =
\begin{cases}
-1 \,, & y \in s \,, \\
+1 \,, & y \notin s \,,
\end{cases}
\end{equation}

where $y$ is a point in the fiber of $\cP$ over $x\in C$
that one blows up in the
process of modification. Note there are $(q+1)$ choices for $y$ since
we assume $\deg x = 1$.

The bundles $\cP_k$ for $k>0$ are distinguished by having a unique section

\begin{equation}
s_\infty: C= \bP\left(\cO(k) \right) \xrightarrow{\quad \quad}
\bP\left(\cO \oplus \cO(k) \right) =
\cP_k\label{eq:193}
\end{equation}

with a negative self-intersection $(s_\infty,s_\infty) = - k$.
According to  (<ref>),
the corresponding $(q+1)$ neighbors of $\cP_k$ look
like this:

\begin{gather}
\raisebox{-2.25cm}{\begin{tikzpicture}[scale=0.9]
\node (0) at (0,0) [circle,draw,minimum width = 1.3cm,fill=gray!3] {$0$};
\node (1) at (2,0) [circle,draw,minimum width = 1.3cm,fill=gray!7] {$1$};
\node (2) at (1.246979604, 1.563662965) [circle,draw,minimum width =
1.3cm,fill=gray!7] {$1$};
\node (3) at (1.246979604, -1.563662965) [circle,draw,minimum width =
1.3cm,fill=gray!7] {$1$};
\node (4) at (-0.4450418670, 1.949855825) [circle,draw,minimum width =
1.3cm,fill=gray!7] {$1$};
\node (5) at (-0.4450418670, -1.949855825) [circle,draw,minimum width =
1.3cm,fill=gray!7] {$1$};
\node (6) at (-1.801937736, 0.8677674786) [circle,minimum width =
1.3cm] {};
\node (7) at (-1.801937736, -0.8677674786) [circle,minimum width =
1.3cm] {};
\node (8) at (-2, 0) [circle,minimum width =
1.3cm] {$\dots$};
\draw (0)--(1);
\draw (0)--(2);
\draw (0)--(3);
\draw (0)--(4);
\draw (0)--(5);
\draw (0)--(6);
\draw (0)--(7);
\draw (0)--(8);
\end{tikzpicture}}\,, \quad
\raisebox{-2.25cm}{\begin{tikzpicture}[scale=0.9]
\node (0) at (0,0) [circle,draw,minimum width = 1.3cm,fill=gray!40] {$k$};
\node (1) at (2,0) [circle,draw,minimum width = 1.3cm,fill=gray!56] {$k+1$};
\node (2) at (1.246979604, 1.563662965) [circle,draw,minimum width =
1.3cm,fill=gray!32] {$k-1$};
\node (3) at (1.246979604, -1.563662965) [circle,draw,minimum width =
1.3cm,fill=gray!32] {$k-1$};
\node (4) at (-0.4450418670, 1.949855825) [circle,draw,minimum width =
1.3cm,fill=gray!32] {$k-1$};
\node (5) at (-0.4450418670, -1.949855825) [circle,draw,minimum width =
1.3cm,fill=gray!32] {$k-1$};
\node (6) at (-1.801937736, 0.8677674786) [circle,minimum width =
1.3cm] {};
\node (7) at (-1.801937736, -0.8677674786) [circle,minimum width =
1.3cm] {};
\node (8) at (-2, 0) [circle,minimum width =
1.3cm] {$\dots$};
\draw (0)--(1);
\draw (0)--(2);
\draw (0)--(3);
\draw (0)--(4);
\draw (0)--(5);
\draw (0)--(6);
\draw (0)--(7);
\draw (0)--(8);
\end{tikzpicture}}\,, \quad k>0 \,, \label{kapusta} \\
\textup{\sl The neighborhood of $\cP_k$ in $\Gr_\bG$.} \notag
\end{gather}

where abbreviate $\cP_k$ to just $k$. Note that $\cP_{-k} \cong
\cP_{k}$; with this in mind one does not need to distinguish between
the two cases
in (<ref>).

The bundles $\cP_k$ represents the the $\bG(\Bbbk[t^{-1}])$-orbits
on $\Gr_{\bG}$ for $\bG=PGL(2)$. The local rules (<ref>)
give a unique reconstruction of those up to an automorphism of the
tree, see Figure (<ref>).

\begin{gather}
\label{tree2}
\raisebox{-100pt}{\includegraphics[height=200pt]{Tree2.pdf}} \\
\textup{\sl Orbits of $\bG(\Bbbk[t^{-1}])$ in $\Gr_\bG$.} \notag
\end{gather}

The grayscale in (<ref>) encodes the isomorphism type as

\begin{equation}
\label{eq:162}
\raisebox{-0.6cm}{\begin{tikzpicture}
\selectcolormodel{gray}
\node (0) [circle,draw,minimum width = 1.3cm,fill=gray!3]  at
(0,0) {$\cP_0$};
\node (1)  [circle,draw,minimum width = 1.3cm,fill=gray!10]  at (2,0) {$\cP_1$};
\node (2)  [circle,draw,minimum width = 1.3cm,fill=gray!17]  at (4,0) {$\cP_2$};
\node (3)  [circle,draw,minimum width = 1.3cm,fill=gray!23]  at (6,0)
\node (4)  [circle,draw,minimum width = 1.3cm,fill=gray!31] at (8,0)
\node (5) [circle,draw,minimum width = 1.3cm,fill=gray!40]  at (10,0)
\node (6) [circle,minimum width = 1.3cm]  at (12,0)
\draw (0)--(1); \draw (1)--(2); \draw (2)--(3);  \draw (3)--(4);
\draw (4)--(5); \draw (5)--(6);
\end{tikzpicture}}
\end{equation}

Any chain of the form (<ref>)  is a
fundamental domain for $PSL(2,\Bbbk[t^{-1}])$. Three such domain
are plotted in (<ref>).  They differ by shifts by element of the

\begin{equation}
\Bbbk^{\infty} \cong \left\{
\begin{pmatrix}
1 & \Bbbk[t^{-1}] \\
0 & 1
\end{pmatrix} \right\} \subset
PSL(2,\Bbbk[t^{-1}])\,, \label{eq:163}
\end{equation}

which fixes the corresponding cusp.

One may compare and
contrast this action with its Archimedean analog, namely,  the action
of $PSL(2,\Z)$ on the hyperbolic plane.
The subgroup (<ref>) is the nonarchimedian analog of

\[
\left\{
\begin{pmatrix}
1 & \Z \\
0 & 1
\end{pmatrix} \right\} \subset
PSL(2,\Z) \,.
\]

Together with $PSL(2,\Bbbk)$, the subgroup (<ref>) generates $PSL(2,\Bbbk[t^{-1}])$.

Summarizing the above discussion, one can say that for $\bG = PGL(2)$,
the action of the
spherical  Hecke algebra on functions on $\Bun_\bG$ amounts to the action
of one operator $\Delta$ on functions $f(k)$ of one variable $k \in \Z_{\ge
0}$ given by the formula

\begin{equation}
\label{eq:173}
\left[\Delta f\right] (k) = q f(k-1) + f(k+1) \,, \quad f(-1) = f(1)
\,.
\end{equation}

This operator is self adjoint with respect to the natural Hermitian
product in $L^2(\Bun_\bG)$ given by

\begin{equation}
\label{eq:174}
\| f \|_{L^2(\Bun_\bG)} = \sum_{k \ge 0}
\frac{|f(k)|^2}{|\Aut(\cP_k)|}  \,,
\end{equation}

which is elementary to see directly using

\begin{equation}
\label{eq:166}
|\Aut(\cP_k)| =
\begin{cases}
q (q^2-1) \,, &  k =0 \,, \\
q^{k+1} (q-1) \,, & k > 0 \,.
\end{cases}
\end{equation}

§.§ Spectrum of $\Delta$

The operator (<ref>) is a second-order
difference operators with constant coefficients. Ignoring the
boundary conditions, we find two eigenfunctions

\[
f_{\pm}(k;a) = \left(q^{1/2} a^{\pm 1} \right)^k, \quad k \in \Z, a
\in \Ct\,,
\]

with eigenvalue

\begin{equation}
\Delta f_{\pm}(k;a) = \lambda(a) \, f_{\pm}(k;a)
\,, \quad \lambda(a) = q^{1/2} (a + a^{-1})  \,. \label{eq:182}
\end{equation}

The boundary condition in (<ref>) singles out one
particular linear combination of those

\begin{equation}
\label{eq:175}
\Eis(k,a) = \left(q^{1/2} a\right)^k + \frac{q a^2 -1}{a^2 - q} \,
\left(q^{1/2} a^{-1} \right)^k\,,
\end{equation}

which is the Eisenstein series for $\Bun_{PSL(2)}(\bP^1)$, see below.

By construction, $\Eis(k,a)$ and $\Eis(k,a^{-1})$ are proportional. In
fact, the elementary relation

\begin{equation}
\label{eq:180}
\Eis(k,a^{-1}) = \frac {a^2 - q}{q a^2 -1} \, \Eis(k,a)
\end{equation}

is the simplest instance of the general functional equation for
Eisenstein series.

In the spectral theory of differential and difference operators,
one views (<ref>) as a superposition of a wave and its
reflection by the boundary. One reads off properties of the
system, such as its spectrum,  from the properties of the
coefficient in (<ref>) as a function of $a$, see e.g.<cit.>. This analysis is
elementary in the case at hand.

Note that none of the functions (<ref>) lies in $L^2$. Indeed,
for large $k$ it grows at least as $q^{k/2}$, making the sum
(<ref>) diverge. Also note that

\[
\Res_{a= \pm \sqrt{q}} \Eis(k,a) \, \frac{da}{a}
= \frac{q-q^{-1}}{2 } \, (\pm
\]

are $L^2$-eigenfunctions with eigenvalues $\pm (q+1)$. As we will
see momentarily

\begin{equation}
\label{eq:176}
\Spec(\Delta) = \big[-2\sqrt{q},2 \sqrt{q}\big] \cup \{\pm (q+1)\} \,,
\end{equation}

with multiplicity one. The spectral measure on the continuos spectrum in (<ref>)  is in the Lebesgue measure
equivalence class.

One popular starting point for spectral decompositions is the

\begin{equation}
f = \frac{1}{2\pi i} \oint_{|z|\gg 1} dz \, \frac{1}{\Delta-z}
\, f\label{eq:165}
\end{equation}

valid for any function or vector $f$ and a bounded linear operator
$\Delta$. The operator $(\Delta-z)^{-1}$ in (<ref>) is called the
resolvent of the operator $\Delta$. It is analytic for $|z| \gg 1$ and
its residue at $z=\infty$ is the identity operator, whence

If $\Delta$ is self-adjoint then by the abstract spectral theorem
one has the following integral representation for the matrix
coefficients of the resolvent

\begin{equation}
\label{eq:321-2}
\left( (\Delta -z )^{-1} f_1, f_2\right) = \int_\R \frac{f_1(\lambda)
\overline{f_2(\lambda)}}{\lambda - z} \,
\mu_{\textup{spectral}}(d\lambda) \,.
\end{equation}

Here we identify elements $f_i$ of our Hilbert space with
functions[in principle, vector-valued, but scalar-valued in our case] on the spectrum of $\Delta$ and
$\mu_{\textup{spectral}}$ is the spectral
measure. It is clear that the atoms of $\mu_{\textup{spectral}}$
produce poles in (<ref>), while the absolutely continuous
part of $\mu_{\textup{spectral}}$ produces a jump across the real
axis in (<ref>). The magnitude of this jump is proportional to the
density of $\mu_{\textup{spectral}}$. A concrete spectral decomposition can thus
be obtained by moving the contour of integration in (<ref>) so
that to make it encircle the singularities of $(\Delta -z )^{-1}$.

A good set of functions $f$ to plug into (<ref>) is not the
Eisenstein series themselves, but their linear combinations of the

\begin{align}
\label{eq:177}
\Eis_\omega(k) &= \oint_{|a|\gg 1} \Eis(k;a) \, \omega(a) \, \frac{da}{2\pi
i a}  \,, \\
&= \oint_{|a|=1} \Eis(k;a) \, \omega(a) \, \frac{da}{2\pi
i a} + \frac{q-q^{-1}}{2} \sum_{\pm } \omega(\pm
q^{1/2}) \, (\pm 1)^k \,, \label{eq:177bis}
\end{align}

where $\omega(a) \in \C[a^{\pm 1}]$.
These are
called pseudo-Eisenstein series in the theory of automorphic forms.
In other situations, they may be called wave packets, etc., that is,
linear combinations of eigenfunctions that decay at infinity sufficiently

Since (<ref>) picks out the coefficients of the $a\to \infty$
expansion of $\Eis(k;a)$, it is easy to see that each $\Eis_\omega(k)$ is
compactly supported and that the map

\begin{equation}
\label{eq:186}
\C[a^{\pm 1}] \xrightarrow{\quad \omega \mapsto \Eis_\omega
\quad} \, \,
\begin{matrix}
\textup{compactly supported} \\
\textup{functions on $\Bun_\bG$}
\end{matrix}
\end{equation}

is surjective. The right hand side is dense in $L^2(\Bun_\bG)$ and
a very useful perspective on the spectral decomposition is to describe
$L^2(\Bun_\bG)$ as the completion of $\C[a^{\pm 1}]$ with respect to
the seminorm  $\| \Eis_\omega \|_{L^2(\Bun_\bG)}$ (which includes
quotienting out by the seminorm's kernel). This seminorm
can be computed
abstractly in great generality, see Section <ref>.

By construction

\begin{equation}
\label{eq:178}
\frac{1}{\Delta-z}
\, \Eis_\omega =
\oint_{|z| \gg |a| \gg 1}
\frac{da}{2\pi i a}
\,
\frac{\omega(a)}{
\lambda(a)- z} \, \Eis(a) \,.
\end{equation}

Moving $z$ away from $z=\infty$ means shrinking the contour of
integration in (<ref>) to the unit circle $|a|=1$ as in
Note the function $\lambda(a)$ is one-to-one for $|a|>1$ and
and two-to-one for $|a|=1$, creating the expected discontinuity of the
resolvent along

\[
\lambda(\{|a|=1\}) = \big[-2\sqrt{q},2 \sqrt{q}\big]  \, .
\]

This proves (<ref>).

For $\bG=SL(2)$, we need to consider only even $k$ and the operator
$\Delta^2$. This means we will have $b=a^2$ as the new parameter
for Eisenstein series

\begin{equation}
\label{EisSL2}
\Eis_{SL(2)}(\cO(m)\oplus \cO(-m),b) = \left(q b\right)^m + \frac{q b -1}{b - q} \,
\left(q b^{-1} \right)^m\,.
\end{equation}

In particular, the eigenfunctions $(\pm 1)^k$ get identified, while
the continuous spectrum retains its shape.

§.§ Same spectrum, as seen in the dual group

The interpretation of this spectrum from the point of the general theory
is the following. One considers the Langlands dual group $\bG^\vee$
and identifies, following Satake,

\[
\lambda: \SH_x  (\bG) \otimes_\Z \Z[q^{\pm 1/2}]
\xrightarrow{\quad \sim \quad} K_{\bG^\vee} (\pt) \otimes_\Z
\Z[q^{\pm 1/2}]  \,,
\]

where the isomorphism is an isomorphism of algebras. In particular,

\[
\]

and vice versa, and

\begin{equation}
\label{eq:181}
\lambda ( \Delta) = q^{1/2} \otimes \textup{defining representation
of $SL(2)$} \,.
\end{equation}

To find an agreement with (<ref>) above, we observe that

\[
\lambda(a) = q^{1/2} \tr
\begin{pmatrix}
a \\
& a^{-1}
\end{pmatrix}
\]

is the trace in the defining representation of $a$, viewed as a
conjugacy class in $SL(2)$. Viewed from this angle,

\begin{equation}
\label{eq:183}
\Spec(\SH_x  (\bG)) \subset \textup{conjugacy classes in
\end{equation}

We hope the reader will enjoy checking that for both $PGL(2)$ and
$SL(2)$ the spectrum computed above can be described as

\begin{equation}
\label{eq:184}
\Spec(\SH_x  (\bG)) = \bigsqcup_{\pi: SL(2,\C)\to \bG^\vee(\C)}
\pi \left( \begin{pmatrix}
q^{1/2} \\
& q^{-1/2}
\end{pmatrix} \right) \cdot \bG^\vee(\C)^\pi_{\textup{compact}}
\end{equation}

where the union is over all homomorphisms
$\pi$, the group $\bG^\vee(\C)^\pi$ is the centralizer of the image of
$\pi$, and $\bG^\vee(\C)^\pi_{\textup{compact}}$ is its maximal
compact subgroup.

Indeed, in the case at hand, there are only two possibilities for $\pi$. We have the
trivial representation, which gives

\begin{equation}
\label{eq:185b}
\Spec(\SH_x  (PGL(2)))_{\textup{continous}} = \left\{
\begin{pmatrix}
a \\
& a^{-1}
\end{pmatrix} , |a|=1 \right\} \,,
\end{equation}

and the defining
representation which gives

\begin{equation}
\label{eq:185}
\Spec(\SH_x  (PGL(2)))_{\textup{discrete}} = \left\{
\pm  \begin{pmatrix}
q^{1/2} \\
& q^{-1/2}
\end{pmatrix}  \right\} \,.
\end{equation}

For $\bG = SL(2)$, one takes the image of (<ref>) and
(<ref>) in $PGL(2,\C)$.

Formula (<ref>) admits extremely far-reaching generalizations of the

\begin{equation}
\label{eq:184bb}
\Spec(\SH_x  (\bG)) = \bigsqcup_{\pi:
\Gamma \times SL(2,\C)\to \bG^\vee(\C)}
\pi \left( \Fr_x \times \begin{pmatrix}
\| \varpi_x \|^{1/2} \\
& \| \varpi_x \|^{-1/2}
\end{pmatrix} \right) \cdot \bG^\vee(\C)^\pi_{\textup{compact}} \,,
\end{equation}

where the spectrum is the joint spectrum of the spherical Hecke
algebras for all nonarchimedian
places of a global field $\F$, $\Gamma$ is some
version of the unramified Galois group of $\F$, $\Fr_x$ is the
conjugacy class of the Frobenius element at $x$, and
$\| \varpi_x \|$ is the norm of a uniformizing element of $\F_x$.

§.§ Eisenstein series

The space $\Bun_\bG$ depends covariantly on the group $\bG$, meaning
that a group homomorphism $\bG \to \bG'$ induces a map $\Bun_\bG \to
\Bun_{\bG'}$. In particular, let $\sP \to \bG$ be an inclusion of a
parabolic subgroup and let $\sP \to \bM$ be the quotient by the unipotent
radical. Both Eisenstein and pseudo-Eisenstein series on $\Bun_\bG$
are produced by the
pull-push $\pi_{\bG,*} \pi_\bM^*$ in the diagram

\begin{equation}
\label{eq:187}
\xymatrix{
& \Bun_{\sP}(C) \ar[ld]_{\pi_\bM} \ar[rd]^{\pi_\bG}\\
\Bun_\bM(C) && \Bun_\bG(C) \,,}
\end{equation}

which appears in many places in our narrative.
Here we have discrete sets with the canonical measure $|\Aut|^{-1}$,
which provides a natural bijection between functions and measures. The
pushforward of a functions is really the pushforward of the
corresponding measure, in other words

\begin{equation}
\label{eq:197}
\big[\pi_* f \big](x) = \sum_{x' \in \pi^{-1}(x)}
\frac{|\Aut(x\phantom{{}'})|}{|\Aut(x')|} \, f(x') \,.
\end{equation}

By construction, $\pi_*$ and $\pi^*$ are transpose with respect to the
natural duality between functions and measures.

In our running example, we have

\begin{equation}
\label{eq:179}
\bM \cong GL(1) = \{ z\} \xleftarrow{\quad}
\sP = \left\{
\begin{pmatrix}
z & * \\
0 & 1
\end{pmatrix}
\right\}
\xrightarrow{\quad}
\bG = PGL(2) \,.
\end{equation}

A $\sP$-bundle over a curve $C$ is an extension

\begin{equation}
\label{eq:198}
0 \to \cM \to \cV \to \cO_C \to 0 \,, \quad \cM \in \Bun_{GL(1)}
= \Pic(C)(\Bbbk) \,.
\end{equation}

Up to isomorphism, it is uniquely encoded by the $\bP^1$-bundle
$\cP = \bP(\cV)$ with a section

\[
s_0: C = \bP(\cM) \to \cP \,.
\]

We have $\cM=s_0^*(\cO_\cP(-1))$ and, in particular,

\begin{equation}
\deg s_0 = - \deg \cM = (s_0,s_0) \,.\label{eq:199}
\end{equation}

The numerator in (<ref>) means that, in the
corresponding count, we do not identify sections that
differ by the automorphism of $\cP$.
In other
words, every one of the the $\frac{|\Aut(\cP)|}{|\Aut(\cP,s)|}$ many
point in each $\Aut(\cP)$-orbit in

\begin{equation}
\label{eq:188}
\left[\pi_{\bG,*} \pi_\bM^* \, \phi \right] (\cP) =
\sum_{\textup{sections $s: C\to \cP$}} \phi(s^* \cO(-1)) \,,
\end{equation}

contributes the same value $\phi(s^* \cO_{\bP^1}(1))$. Here $\phi$ is an
arbitrary function on

\begin{equation}
\Pic(C)(\Bbbk) = \Z \oplus \Pic_0(\C)(\Bbbk) \label{eq:200} \,,
\end{equation}

where the $\Z$-coordinate is given by (<ref>).
In our running example $C=\bP^1$, the second summand in (<ref>) is zero.

As we will see momentarily, choosing $\phi(\deg s)$ to be compactly
supported gives pseudo-Eisenstein series on
$\Bun_{PGL(2)}$. Equivalently, we will check that taking

\begin{equation}
\phi(\cM) = \left(q^{1/2} a\right)^{\deg \cM} \label{eq:191}
\end{equation}

recovers (<ref>), which means that

\begin{equation}
\label{eq:192}
\Eis(\cP,a) =
\sum_{\textup{sections $s: C\to \cP$}} \left(q^{1/2}
a\right)^{-\deg(s)}  \,.
\end{equation}

The shift by $q^{1/2}$ is the shift by
$q^{\rho}$ where $\rho=1/2$ is the half-sum of weights of $\bM$ in
the unipotent radical of $\sP$.

It is obvious from (<ref>) that the right-hand side of
(<ref>) is
an eigenfunctions of $\Delta$ with eigenvalue $\lambda(a)$.
Further, the normalization is correct since $\cP_k$ contains
a unique section (<ref>) of degree $-k$. Thus, (<ref>)
is established.

Nonetheless, it will in instructive to do the actual counting.
Actually, we will do
explicitly a middle-school version of the counting we
did just slightly more abstractly in Section <ref>.

§.§ Counting quasisections

have $\cP_k \setminus s_\infty \cong \cO_{\bP^1}(k)$ and hence
all other sections can be identified with rational sections of
$\cO(k)$. In other words, they have the form

\[
s(t) = \frac{F_{d+k}(t)}{G_{d}(t)}, \quad d=0,1,2,\dots\,,
\]

where $F_{d+k}$ and $G_{d}$ are polynomials of the indicated
degree. The degree of this section is $k+2d$.

One way to count these, is to first count pairs $(F_{d+k},G_{d})$
with $G_d \ne 0$ up to an overall constant
multiple, ignoring the fact that they may have a common nonconstant
factor. The corresponding pairs $(F_{d+k}(t),G_{d}(t))$ describe
quasisections of $\cO_{\bP^1}(k)$, see Section <ref>. They
form a complement to a linear subspace in a projective space of dimension
$2d+k+1$. Hence we get the count

\begin{equation}
\label{eq:194}
\sum_{d\ge 0} \left(q^{1/2}
a\right)^{-k-2d} \frac{q^{2d+k+2}-q^{d+k+1}}{q-1} =
\frac{q}{(1-q a^{-2})(1-a^{-2})} \, \left(q^{1/2}
a^{-1}\right)^{k} \,.
\end{equation}

For the common multiple up to a constant factor, we similarly get the

\begin{equation}
\label{eq:194b}
\sum_{d\ge 0} \left(q^{1/2}
a\right)^{-2d} \frac{q^{d+1}-1}{q-1} =
\frac{1}{(1-a^{-2})(1-q^{-1}a^{-2})} \,.
\end{equation}

By definition, counting nonzero polynomials up to a constant factor is
the same as counting closed points of $\bP^1$. This explains why the $\zeta$-function

\begin{equation}
\label{eq:195}
\zeta_{\bP^1}(t) = \frac{1}{(1-t)(1-qt)}
\end{equation}

appears in the answer. Introducing the symmetrized version of (<ref>),
also known as the completed $\zeta$-function

\begin{equation}
\label{eq:195-2}
\xi_{\bP^1}(t) = \frac{q^{1/2} t}{(1-t)(1-qt)} \,,
\end{equation}

we can write the ratio of (<ref>) and (<ref>) as

\begin{equation}
\label{eq:196}
\Eis(\cP_k, a) = \left(q^{1/2}
a\right)^{k} + \frac{\xi(a^{-2})}{\xi(a^2)} \left(q^{1/2}
a^{-1}\right)^{k} \,.
\end{equation}

Eisenstein series for $C=\bP^1$ are proportional to the their constant
terms, see (<ref>) below. The formula (<ref>) is thus
equivalent to the simplest instance of Langlands' constant term

§.§ Constant term and $L^2$-norms

By definition, the constant term

\[
\CT = \pi_{\bM,*} \pi_{\bG}^*
\]

is the transposed operator in the diagram (<ref>).
For $C = \bP^1$ and $\deg \cM \ge -1$, any extension in (<ref>)
is trivial, thus

\begin{equation}
\CT (\phi) (\cM) = \frac{1}{q^{\deg \cM +1}} \, \phi( \cM
\oplus \cO) \,, \quad \deg \cM \ge -1 \,. \label{eq:201}
\end{equation}

The prefactor in (<ref>) comes from the ratio

\[
\frac {|\Aut (\cM)|} {|\Aut ( \cM \subset \cM\oplus\cO)|}=
\frac1{|H^0(\cM)|} = \frac{1}{q^{\deg \cM +1}} \,.
\]

in formula (<ref>).

\begin{equation}
\CT(\Eis(a))(\cO(k)) = \frac1q \left[\left(q^{-1/2}
a\right)^{k} + \frac{\xi(a^{-2})}{\xi(a^2)} \left(q^{-1/2}
a^{-1}\right)^{k} \right]\label{eq:189} \,.
\end{equation}

Langlands gave a very general formulation and very general proof of
such constant term formulas in <cit.>, further investigated by Harder
in <cit.> and many other authors.

For a general curve $C$ and Eisenstein series associated
to $\bG/\bB$, where $\bG$ is a split reductive group and $\bB\subset
\bG$ is the Borel subgroup, formula (<ref>) generalizes as follows. In this case
$\bM$ is the maximal torus of $\bG$, hence

\[
\Bun_\bM = \Cochars(\bM) \otimes_\Z \Pic(C)
\xrightarrow{\quad \deg \quad} \Cochars(\bM) \,.
\]

The variable $a$ in the formula

\begin{equation}
\phi(\cM) = (q^{\rho} a)^{\deg \cM} \label{eq:198-2}
\end{equation}

should thus be interpreted
as an element of the dual torus

\[
a \in \bM^\vee  = \Chars(\bM) \otimes_\Z \Ct \,.
\]

The Weyl group $W$ acts on functions of $a$, such as (<ref>)
in the natural way. Consider the operator

\begin{align}
\boldsymbol{\Sigma} & = \sum_{w\in W} \prod_{
\substack{\alpha > 0 \\ w^{-1} \alpha < 0}}
\frac{\xi(a^{-\alpha})}
{\xi(a^{\alpha})} \,   w \label{eq:199-2} \,, \\
& =  \boldsymbol{\Pi}^{-1} \left(\sum_{w\in W}  w
\right) \boldsymbol{\Pi}
\label{eq:199bis}
\end{align}

where $\boldsymbol{\Pi} = \prod_{\alpha > 0} \xi(a^{-\alpha})$. It
follows from (<ref>) that

\begin{equation}
\label{eq:190}
\boldsymbol{\Sigma}^2  = |W| \boldsymbol{\Sigma} \,,
\end{equation}

which means that $\Sigma$ is proportional to a projector when
acting on rational functions of $a$. With these notations,

\begin{equation}
\label{eq:197-2}
\CT (\Eis(\, \cdot \, )) \,  \propto \, \boldsymbol{\Sigma}
\end{equation}

when acting on functions of the form (<ref>). For functions
$\phi$ that involve a nontrivial character of $\Pic_0(C)(\Bbbk)$, the function
$\xi$ in (<ref>) should be replaced by the corresponding
Dirichlet L-function.

One immediate application of constant term formulas is a
formula of the norm $\| \Eis_\omega \|_{L^2(\Bun_\bG)}$ of a
pseudo-Eisenstein series. This is because we can
we can express norms on $\Bun_\bG$ in terms of norms on $\Bun_\bM$
using the relation

\begin{equation}
\left(\pi_{\bG,*} \pi_\bM^* \right)^* = \CT. \label{eq:11}
\end{equation}

The space $\Bun_\bM$ for $\bM=GL(1)$ is just $\Z$, each point of which has
the automorphism group $GL(1,\Bbbk)$ of order $q-1$. We can
go between functions

\[
\omhat: \Z \to \C
\]

and functions $\omega(a)$ on $\C^\times = \bM^\vee$ by means of the Fourier
transform, which we normalize by

\begin{equation}
\omega(a)  =\sum_k \omhat(k) \,  (q^{1/2} a)^{-k} \,.\label{eq:9}
\end{equation}

The shift in (<ref>) is introduced so that there is no shift

\begin{equation}
\Eis_\omega =\pi_{\bG,*} \pi_\bM^* \,\,  \omhat(\deg(\, \cdot \,))\,.  \label{eq:203}
\end{equation}

We have

\begin{equation}
\| \omhat \|_{L^2(\Bun_\bM)} = \tfrac{1}{q-1} \sum_{k\in \Z}
|\omhat(k)|^2 \,\label{eq:10} \,,
\end{equation}

where the prefactor accounts for the automorphisms.

Formulas (<ref>) and (<ref>) imply

\begin{equation}
\label{eq:205}
\Big( (q^{-1/2} a)^{\deg}, \omhat\Big)_{L^2(\Bun_\bM)} =
\tfrac{1}{q-1} \, \bar{\omega}(a^{-1}) \,,
\end{equation}

where $\bar{\omega}(a) = \overline{\omega(\overline{a})}$.
From this, (<ref>) and (<ref>), we conclude

\begin{equation}
\notag
\Big(\Eis(a), \Eis_{\omega}\Big)_{L^2(\Bun_\bG)} = \Big(\!\CT \Eis(a),
\omhat\Big)_{L^2(\Bun_\bM)}
= \tfrac{1}{q(q-1)} \boldsymbol{\Sigma} \, \bar{\omega}(a^{-1})
\,,
\label{eq:204}
\end{equation}

\begin{equation}
\label{eq:206}
\boldsymbol{\Sigma} \, \omega (a) =  \omega(a) +  \frac{\xi(a^{-2})}{\xi(a^{2})}
\omega(a^{-1})  \,.
\end{equation}

It follows that

\begin{equation}
\label{eq:202}
\| \Eis_\omega \|_{L^2(\Bun_\bG)} =
\tfrac1{q(q-1)} \oint_{|a|\gg 1} \frac{da}{2\pi i a}       \,
\omega(a)
\boldsymbol{\Sigma}
\, \bar{\omega}(a^{-1})\,.
\end{equation}

Moving the contour of integration in (<ref>) as in
(<ref>), we obtain

1q(q-1) ∮_|a|=1 da/2πi a
ω(a)       +
r_1 |ω(q^1/2)|^2 + r_2 |ω(-q^1/2)|^2 ,

12q(q-1)       Σ   ω^2_L^2(|a|=1)  _      +
r_1 |ω(q^1/2)|^2 + r_2 |ω(-q^1/2)|^2
_ ,

where the coefficients $r_1,r_2 >0$ are obtained from the residues of
$\boldsymbol{\Sigma}$ at $a=\pm q^{1/2}$, and where in (<ref>)
we used the fact that $\frac12\boldsymbol{\Sigma}$ is a self-adjoint
projector acting on $L^2(|a|=1)$.

Let us now go back to the discussion in Section <ref>. Formula
(<ref>) describes a seminorm on $\C[a^{\pm 1}]$. Its three terms
involve values of $\omega$ at disjoint closed sets, hence the
completion $L^2(\Bun_\bG)$
splits into a direct sum of two 1-dimensional eigenspaces and
their orthogonal complement. The latter is a direct integral of
Eisenstein series $\Eis(a)$ for $a$ on the unit circle $|a|=1$ modulo
$a\mapsto a^{-1}$. The $a\mapsto a^{-1}$ quotient appears as
the quotient by the kernel of the spectral
projector $\frac12\overline{\boldsymbol{\Sigma}}$. It reflects
the functional equation for $\Eis(a)$.

§.§ A geometric interpretation

To conclude this Appendix, we want to explain our geometric
point of view on spectral decomposition in the
running example. The relevant geometric object is yet another
projective line

\[
\bP^1 = \bG^\vee/\bB^\vee\,,
\]

which is now interpreted as the flag variety for the Langlands
dual group. It has the natural action of

\[
\bM^\vee = \{\diag(a^{-1},a) \} \subset \bG^\vee  \,.
\]

In fact, the group $\bM^\vee$ will play two roles because it both
acts on $\bP^1$ and is also the Levi subgroup of the group
$\bB^\vee$ which appears in the quotient construction of $\bP^1$.
To account for these two roles, one can consider quotient stacks like
$\bM^\vee \backslash \bG^\vee/\bB^\vee$ or even $\bB^\vee \backslash
\bG^\vee/\bB^\vee$, in which $\bM^\vee$ appears  on the left and
on the right on an equal footing.
This results in the

\begin{equation}
\label{eq:12}
K_{\bM^\vee}\!(\bP^1) = K_{\bB^\vee}\!(\bP^1) = K(\bB^\vee
\backslash \bG^\vee/\bB^\vee) = \dots = \Z[\bM^\vee] \boxtimes_{\Z[\bM^\vee]^W}
\Z[\bM^\vee]
\,,
%\quad K(\bM^\vee) = \Z[a^{\pm 1}]\,,
\end{equation}

where dots denote many other possible geometric realizations of
this algebra, the Weyl group $W\cong S(2)$ acts by $a\mapsto a^{\pm
1}$, and the $\boxtimes$ symbol is used to distinguish this
tensor product from the tensor product in equivariant K-theory (which
is the algebra product in (<ref>)). The
identification may be fixed so that

\begin{align*}
\label{eq:13}
a \boxtimes 1 &= \textup{character $a$} \otimes \cO_{\bP^1} \,,\\
1 \boxtimes a & = \cO_{\bP^1}(1) \,.
\end{align*}

We will write elements of $K_{\bM^\vee}\!(\bP^1) \otimes_\Z\C$ in
the form $\omega_1(a) \boxtimes \omega_2(a)$.

There is a distinguished linear functional $\int_{\bP^1}$, given by the horizontal
arrow in the following diagram

\begin{equation}
\label{eq:15}
\xymatrix{
K_{\bM^\vee}\!(\bP^1) \ar[rrr]^{\quad\int_{\bP^1}=(\bM^\vee \backslash \bP^1 \to \pt)_*\quad}\ar[rd]_{\chi}&&& \Z\,, \\
& \Z[\bM^\vee] \ar[rru]_{\textup{\,\, invariants}}
\end{equation}

where $\chi$ is the Euler characteristic, equivalently pushforward to
$\pt/\bM^\vee$. It may be computed by the localization formula in equivariant
K-theory, or the Weyl character formula for $SL(2)$, or the formula
for the sum of a geometric progression.  The result is

\begin{equation}
\label{eq:16}
\int_{\bP^1} \omega_1(a) \boxtimes \omega_2(a) =
\int_{|a|=1} \frac{da}{2\pi i a} \, \omega_1(a) \left( \sum_{w \in W} w \right)
\frac{ \omega_2(a)}{1-a^{-2}}  \,.
\end{equation}

Since the symmetrization eliminates the poles in
(<ref>), the integration contour $|a|=1$
may be moved arbitrarily.

Consider the cotangent bundle

\[
\bX^\vee = T^*\bP^1 = T^*(\bG^\vee/\bB^\vee)\,.
\]

Using the identification

\[
K_{\bM^\vee}\!(\bP^1) = K_{\bM^\vee}\!(\bX^\vee) \,,
\]

one may similarly consider pushing forward along $\bX^\vee \to \pt$.
Since this pushforward is not proper, a certain care needs to be
exercised. Concretely, let $\bfL \subset \bX^\vee$ be the set of
points that have a limit under the action of $a$ as $a\to 0$. This is
the union of the zero section $\bP^1 \subset T^*\bP^1$ and the cotangent
fiber over one of the fixed
points. Let $\chi_\bfL$ denote Euler characteristic with support in
$\bfL$. It takes values in formal Laurent series in $a^{-1}$.
Consider the diagram

\begin{equation}
\label{eq:15-2}
\xymatrix{
K_{\Ct_q \times \bM^\vee}\!(\bX^\vee)
\ar[rrr]^{\quad\int_{\bX^{\!\vee}}\quad}\ar[rd]_{\chi_\bfL}&&& \Z[q^{\pm 1}]\,, \\
& \Z[q^{\pm1}][a,a^{-1}]\!] \ar[rru]_{\textup{\,\, invariants}}
\end{equation}

in which we enlarged the equivariance by making $\Ct$ with coordinate
$q$ act by scaling the cotangent directions with weight $q^{-1}$.
Again, by localization or geometric series, we compute

\begin{equation}
\label{eq:16-2}
\int_{\bX^{\!\vee}} \omega_1(a) \boxtimes \omega_2(a) =
\int_{|a|\gg 1} \frac{da}{2\pi i a} \, \omega_1(a)
\left( \sum_{w \in W} w \right)
\frac{\omega_2(a)}{(1-a^{-2})(1-q a^{2})}  \,.
\end{equation}

Note the similarity between the weight here and the $\zeta$-function
(<ref>) of $\bP^1$.

Now consider the following  Hamiltonian reduction of $\bX^\vee$

\[
\cT = T^*(\bB^\vee \backslash \bG^\vee/\bB^\vee)  = \bB^\vee
\backslash \mu_{\bB^\vee}^{\!-1}(0) \,,
\]

\[
\mu_{\bB^\vee}: \bX^\vee \to \Lie(\bB^\vee)^*\,,
\]

is the moment map for the action of $\bB^\vee$. The
map $\mu_{\bB^\vee}$
is equivariant for the action of $\Ct_q \times \bM^\vee$ provided
$\Ct_q$ scales the target with weight $q^{-1}$. Again, we have
the identification

\[
K_{\bM^\vee}\!(\bP^1) = K(\cT) \,,
\]

and we would like to define a linear functional $\int_\cT$ as before.
Since the map $\cT \to \pt$ is not proper, this does require an
explanation, but the explanation is easier than the one for

We can choose $\bB^\vee\supset \bM^\vee$ so that $\bM^\vee$ acts
with weight $a^2$ on its unipotent radical $\bU^\vee\subset \bB^\vee$. Importantly, the points
in the unipotent radical have a limit as $a\to 0$. This means that
one of the two equations in $\mu_{\bB^\vee}=0$ defines
the Lagrangian $\bfL$. Therefore, the Euler characteristic
is well-defined as an element of

The equation $\mu_{\bB^\vee}=0$ cuts the Euler class of a trivial
bundle with a nontrivial group action. Similarly, the
Chevalley-Eilenberg complex for $\bU^\vee$-invariants cuts out
the Euler class of a trivial line bundle with a nontrivial weight.
This gives

\begin{equation}
\label{eq:17}
\int_{\cT} \omega_1(a) \boxtimes \omega_2(a)  =
\int_{\bX^{\!\vee}} \cE \omega_1(a) \boxtimes \omega_2(a)  \,,
\end{equation}

where $\cE$ is the Euler class

\[
\cE = \underbrace{(1-q)(1-q a^2)}_{\mu_{\bB^\vee}=0}
\underbrace{(1-a^{-2})}_{\textup{$\bU^\vee$-invariants}} \,.
\]

This means

\begin{equation}
\label{eq:16-3}
\int_{\cT} \omega_1(a) \boxtimes \omega_2(a) =
\int_{|a|\gg 1} \frac{da}{2\pi i a} \,
\left(\omega_1(a) \omega_2(a) + \frac{\xi(a^{-2})}{\xi(a^{2})} \,
\omega_1(a) \omega_2(a^{-1})\right)\,,
\end{equation}

Comparing this with (<ref>), we see that

\begin{equation}
\label{eq:18}
\left( \Eis_{\omega_1}, \Eis_{\omega_2}\right)_{L^2(\Bun_\bG)}
\propto \int_{\cT} \omega_1(a) \boxtimes \omega^*_2(a)\,,
\end{equation}

\begin{equation}
\label{eq:19}
\omega^*(a) = \bar{\omega}(a^{-1}) \,.
\end{equation}

The proportionality constant in (<ref>) is  interesting
and has to do with volumes of $\Bun_\bG$ and other things. It is, however, immaterial
for the spectral decomposition.

Now, how does one see the spectral decomposition and specifically
(<ref>) from this ? For any reductive group $\bG^\vee$, the moment

\begin{equation}
\label{eq:20}
\mu_{\bG^\vee}: T^*(\bG^\vee/\bB^\vee)  \to \Lie(\bG^\vee)^* \cong
\Lie(\bG^\vee)
\end{equation}

is the resolution of the cone of nilpotent elements in
$\Lie(\bG^\vee)$. The fibers of this map are called Springer fibers.
The map (<ref>) induces a map

\begin{equation}
\label{eq:8}
\bmu:  \cT\to
\Lie(\bG^\vee)/\bG^\vee\,,
\end{equation}

the fibers of which are Cartesian squares of the Springer

Consider the diagram

\begin{equation}
\label{eq:21}
\xymatrix{
\cT\big/\Ct_q \ar[rr] \ar[dr]_{\bmu}&& \pt\big/\Ct_q \\
&\Lie(\bG^\vee)\big/(\Ct_q \times \bG^\vee) \ar[ru]} \,.
\end{equation}

The pushforward $\bmu_*$ in (<ref>) gives a K-theory class
on the stack of nilpotent conjugacy
classes $e$ for the dual group $\bG^\vee$. These
correspond to representations $\pi$ in (<ref>)

\begin{equation}
\label{eq:14}
e = \pi\left(\!\left(
\begin{matrix}
0  & 1\\ 0 & 0
\end{matrix} \right)\!\right)  \,.
\end{equation}

We have

\begin{equation}
\label{eq:22}
\textup{Orbit}(e)\big/ (\Ct_q \times \bG^\vee) \cong
\pt \big/  (\Ct_q \times \bC(e)) \,,
\end{equation}

where $\bC(e)$ is the centralizer of $e$ and $q\in \Ct_q$ acts by
$\pi(\diag(q^{1/2},q^{-1/2}))$ in the fiber over $e$. We see
this action of $q$ explicitly in (<ref>). Restricted to
(<ref>), the northeast
going map in (<ref>) is the integration over the maximal
compact subgroup of the complex group $\bC(e)$. This
group coincides with $\bG^\vee(\C)^\pi_{\textup{compact}}$ in
(<ref>). This identifies the spectrum as a set of conjugacy
classes in $\bG^\vee$.

Further analysis of (<ref>)
describes spectral projectors as pushforwards along the Spinger
fibers and computes the spectral measure from the
$\Ct_q \times \bC(e)_{\textup{compact}}$-action on the
unipotent radical of $\bC(e)$. See [5] for many
technical details required for the complete match. One key
property that needs to be checked is that the pieces corresponding
to each $e$ are positive in the same way as
(<ref>) is a sum of orthogonal seminorms. Here the
Hermitian structure is introduced by (<ref>).
It would be
interesting to have an abstract argument showing such positivity.

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