contain some technical results concerning the semi-infinite Bruhat order; in particular, we prove the existence of analogs of Deodhar lifts for the semi-infinite Bruhat order.

Acknowledgments. {#acknowledgments. .unnumbered}
----------------

We thank Michael Finkelberg for sending us unpublished manuscripts. S.K. was partially supported by JSPS Grant-in-Aid for Scientific Research (B) 26287004 and Kyoto University Jung-Mung program. S.N. was partially supported by JSPS Grant-in-Aid for Scientific Research (B) 16H03920. D.S. was partially supported by JSPS Grant-in-Aid for Scientific Research (C) 15K04803.

Algebraic setting. {#sec:notation}
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Affine Lie algebras. {#subsec:liealg}
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A graded vector space is a ${\mathbb{Z}}$-graded vector space over ${\mathbb{C}}$ all of whose homogeneous subspaces are finite-dimensional. Let $V = \bigoplus_{m \in {\mathbb{Z}}} V_{m}$ be a graded vector space with $V_{m}$ its subspace of degree $m$. We define $${\mathop{\rm gdim}\nolimits}V := \sum_{m \in {\mathbb{Z}}} \bigl( \dim V_{m} \bigr) q^{m}.$$ Also, we denote by $V^{\vee}$ (resp., $V^{\ast}$) the full (resp., restricted) dual of $V$; note that $V^{\ast} := \bigoplus _{m \in {\mathbb{Z}}} ( V^{\ast} )_{m}$, with $(V^{\ast})_{m}:=(V_{-m})^{\ast}$. In addition, we set ${\widehat{V}}:=\prod_{m \in {\mathbb{Z}}} V_{m}$, which is a completion of $V$.

Let $G$ be a connected, simply-connected simple algebraic group over ${\mathbb{C}}$, and $B$ a Borel subgroup with unipotent radical $N$. We 