Yang-Baxter basis of Hecke algebra and Casselman's problem (extended abstract)
Maki Nakasuji, Hiroshi Naruse
TL;DR
This work extends the Yang-Baxter basis from type $A$ to arbitrary Lie types by formulating a two-parameter generic Hecke algebra and constructing a Yang-Baxter basis $\{Y_w\}$ with robust duality to the standard basis $\{h_w\}$. A Kostant-Kumar twisted group algebra is employed to realize $Y_w$ via a canonical isomorphism, yielding explicit formulas for transition coefficients and enabling recurrence relations for efficient computation. The authors then apply this framework to Casselman's problem on Iwahori-fixed vectors of principal series representations of $p$-adic groups, obtaining explicit expressions for the interbasis coefficients after a specialization $t_1=-q^{-1}, t_2=1$, and connecting these to Whittaker functions. The work further clarifies the relationship with Bump-Nakasuji’s results, showing consistency and equivalence of conjectures in the simply-laced case, and thereby linking geometric, algebraic, and automorphic perspectives. Overall, the paper provides a unified algebraic approach to transition phenomena in Hecke algebras, principal series intertwiners, and related $K$-theoretic and combinatorial structures, with concrete formulas and applications to Casselman’s problem and BN’s conjectures.
Abstract
We generalize the definition of Yang-Baxter basis of type $A$ Hecke algebra introduced by A.Lascoux, B.Leclerc and J.Y.Thibon (Letters in Math. Phys., 40 (1997), 75--90) to all the Lie types and prove their duality. As an application we give a solution to Casselman's problem on Iwahori fixed vectors of principal series representation of $p$-adic groups.