Shifts in the decomposition of Bott-Samelson bimodules

Let $k$ be an algebraically closed field of characteristic $0$, let $V=k^n$ be a $k$ vector space of dimension $n$, and let $R=k[V]$ be the ring of polynomial functions on $V$. Suppose that $W\subset\operatorname{GL}(V)$ is a finite reflection group, and let $S\subset W$ be a set of simple reflections so that $(W,S)$ is a finite … Read more

Semisimple vs ordinary trace of Frobenius on nearby cycles of affine flag variety

Let G→X be a parahoric group scheme over a curve, with parahoric level structure at x0. Gaitsgory essentially showed that the nearby cycles functor RΨ takes perverse sheaves on the affine Grassmannian GrG|X−x0 to central perverse sheaves on the affine flag variety GrG|x0=FlG. This shows that the Frobenius trace function is central in the parahoric … Read more

Kazhdan-Lusztig basis elements appearing in product with distinguished involution

My apologies if the below is too malformed to make sense. Let (W,S) be the affine Weyl group of a reductive group G, and let {Cw} be the Kazhdan-Lusztig C-basis (an answer in terms of the C′w-basis is of course equally useful, though). Let H be the corresponding Hecke algebra over Z[q,q−1], and write hx,y,z … Read more

Relationship between Hecke algebra and center of universal enveloping algebra (and the Harish-Chandra isomorphism)

Let G be a semisimple Lie group of noncompact type and let K be a maximal compact subgroup. Let g=p⊕k be the Cartan decomposition coming from some Cartan involution, and let a be a maximal abelian subalgera of p. Let A be the subgroup of G corresponding to a. It is known that G can … Read more

On the order of the head of product of two simple modules over Quiver Hecke Algebras

My question is: We assume the underlying quiver is a Dynkin quiver. Let L(λ) and L(μ) be two simple modules over Quiver Hecke algebra R where λ and μ are two Konstant partitions with condition λ≻μ. This order is deduced from one ordering of positive roots. If one of them is real simple module. We … Read more

Index of the Hecke algebra with operators omitted

This is a spin-off to the question Omitting primes from a Hecke algebra by David Loeffler. Let N be a positive integer. For a finite set of primes Σ, let TΣ be the Z-subalgebra of endomorphisms of S2(Γ1(N)) generated by Hecke operators Tℓ for all prime ℓ∉Σ. If ℓ∈Σ implies that ℓ∤, then it is … Read more

A problem on Kazhdan–Lusztig theorem

I am reading Chriss, Ginzburg’s book Representation theory and complex geometry. In theorem 7.2.16 it says that the convolution action of the Steinberg variety $St=\tilde{\mathcal{N}}\times_\mathfrak{g}\tilde{\mathcal{N}}$ on the Springer resolution $\tilde{\mathcal{N}}$ induces an action on $K$-groups, i.e. an action of $K_0^{G\times\mathbb{G}_m}(St)=H_\text{aff}$ on $K_0^{G\times\mathbb{G}_m}(\tilde{\mathcal{N}})=R(T)$. Then the claim is that the element $T_{s_\alpha}\in H_\text{aff}$ acts as $$T_{s_\alpha}:e^\lambda\mapsto\frac{e^\lambda-e^{s_\alpha(\lambda)}}{e^{\alpha}-1}-q\frac{e^\lambda-e^{s_\alpha(\lambda)+\alpha}}{e^{\alpha}-1}$$ where … Read more

Finite-dimensional representations of DAHA

It is shown by Berest-Etingof-Ginzburg that there exist finite-dimensional irreducible representations of rational Cherednik algebra $H_c(S_n)$ of $A_{n-1}$ type if and only if the deformation parameter $c$ takes the rational numbers of the form $c=m/n$. Since the rational Cherednik algebra is the rational degeneration of double affine Hecke algebra (DAHA), the question is as follows: … Read more

Is there a way to define Hecke operators “inherently” as certain endomorphisms of the Jacobian?

From the Eichler-Shimura relation, we have a formula for $T_p$ when we reduce $\textrm{End}(\textrm{Jac}(X))$ mod $p$. Explicity, $T_p=\textrm{Frob}_p+p\textrm{Frob}_p^{-1}$. Is there a way to define the Hecke operator as a lift of this operator satisfying certain other properties? Is there a definition of $T_p$ which does not rely on a moduli space interpretation or double coset … Read more