The geometry of tilting composition series via Richardson varieties

TL;DR

This work provides a geometric interpretation of the graded Jordan-Hölder multiplicities of indecomposable tilting sheaves on flag varieties by identifying them with the rank of the hypercohomology of a motivic geometric extension on Richardson varieties in the Langlands dual flag variety. It unifies mixed tilting theory, modular Koszul duality, Ringel duality, and $\ell$-Kazhdan-Lusztig polynomials to give explicit formulas for the multiplicities in terms of $\ell$-KL data, valid in characteristic $\ell\ge 0$. A central advance is the introduction of the geometric extension $\mathscr{E}(X^z_x)$ on Richardson varieties, whose Frobenius-trace computations reproduce the same multiplicity polynomials and are supported in a single parity. The results extend known characteristic-zero formulas to positive characteristic and illuminate the role of Richardson variety geometry as a natural geometric substrate for tilting multiplicities in geometric representation theory.

Abstract

We prove the (graded) Jordan-Hölder multiplicities of (mixed) tilting sheaves on flag varieties admit a geometric interpretation as the hypercohomology of certain sheaves on Richardson varieties in the Langlands dual flag variety. These sheaves are a motivic variant of geometric extensions, and provide a replacement for parity sheaves on the Richardson variety. We also provide an explicit formula for these multiplicities in terms of $\ell$-Kazhdan-Lusztig polynomials.

Theorems & Definitions (37)

• Theorem 1
• Theorem 2
• Theorem 3
• Remark 4
• Proposition 5
• proof
• Corollary 6
• Remark 7
• Corollary 8
• Lemma 9