Notes on the structure constants of Hecke algebras of induced representations of finite Chevalley groups

Abstract

This paper contains an algorithm for the structure constants of the Hecke algebra of a Gelfand-Graev representation of a finite Chevalley group.

On Refined Bruhat Decompositions and Endomorphism Algebras of Gelfand-Graev Representations

“…BxB ∩ zBy −1 B. Here Curtis also raises the problem of explicitly obtaining formulas for the structure constants, which can be used to give a combinatorial proof of the existence of the homomorphisms f T , see [Cur15,Section 4]. The interest in the above cell intersections and the problem raised by Curtis on an explicit determination of structure constants of H are the motivation lying at the core of our work.…”

“…Gelfand-Graev characters. Following [Cur15], let ψ be a linear representation of U and let e = |U | −1 u∈U ψ(u −1 )u ∈ CG be the idempotent such that the induced representation ψ G is afforded by the module CGe. The Hecke algebra corresponding to ψ is (as in [CR81, §11.D])…”

“…An algorithm for obtaining these constants was provided by C. Curtis (see [Cur15]). A variation of this algorithm, with different choices for U -coset representatives was described in [Sim16].…”

“…In order to obtain a constructive description of each f T , one needs to consider structure constants of H with respect to some basis. Building on work of Kawanaka [Kaw77] and Deodhar [Deo85], Curtis described in [Cur15] an algorithm for obtaining structure constants for H with respect to a standard basis parametrized by certain elements in N G (T ). In general, the determination of the structure constants requires a deep understanding of the following intersections of shifted U -double cosets, U xU ∩ zU y −1 U, for U ∈ Syl p (G) and x, y, z ∈ N G ( T ).…”

Free probability on <inline-formula><tex-math id="M1">\begin{document}$ C^{*}$\end{document}</tex-math></inline-formula>-algebras induced by hecke algebras over primes

“…Here, we are interested in Hecke algebras, themselves, and operator theory on them via representation determined by free probability on the algebras (e.g., see [10], [11], [12] and [13]). More about other researches in Hecke algebra theory, e.g., see [15], [19], [24] and cited papers therein.…”