(e-mail: [sagaki@math.tsukuba.ac.jp]{})
title: |
     **Equivariant $K$-theory of\
    semi-infinite flag manifolds\
    and Pieri-Chevalley formula[^1]**
---

Introduction. {#sec:intro}
=============

Let $G$ be a connected and simply-connected simple algebraic group over ${\mathbb{C}}$, and let $X$ be the flag variety of $G$. The torus-equivariant Grothendieck group $K_H ( X )$ of $X$ affords rich structures from the perspective of geometry and representation theory. One of the highlights there is the positivity of the structure constants of the products among natural classes (called the Schubert classes; see Anderson-Griffeth-Miller [@AGM], and Baldwin-Kumar [@BK]), which serves as a basis of its interaction with the eigenvalue problems [@Kly98] and Gaudin models [@MTV]. There is a variant of this theme (called the Pieri-Chevalley formula), namely the structure constants of the products between Schubert classes and (ample) line bundles in $K_H ( X )$, which is also known to be positive by Mathieu [@Mat00] and Brion [@Br02].

Pittie and Ram [@PR99] initiated a program to describe such a positive structure constant by relating them with the standard monomial theory (SMT for short). In particular, they gave an explicit meaning of each structure coefficient in the Pieri-Chevalley formula in terms of Lakshmibai-Seshadri paths (LS paths for short; see, e.g., [@Lit95]), which 