Questions tagged [affine-hecke-algebras]
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19 questions
9
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1
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Geometric interpretations of nil-Hecke ring and affine Hecke algebra
I am interested in two related constructions which give us either the cohomology or the $T \times \mathbb{C}^*$-equivariant $K$-theory of flag varieties.
Let $G$ be a semisimple, simply connected ...
- 841
8
votes
2
answers
974
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Relation between representations of p-adic groups and affine Hecke algebras
Let $R_n$ be the category of complex-valued smooth finite-length representations of the group $GL_n(F)$, where $F$ is a local field.
By the result of Borel, the subcategory of $R_n$ consisting of ...
- 6,201
8
votes
1
answer
595
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Affine vs Yokonuma
Let $G=GL_n$. Let us start with the Hecke algebra $H_n$. It acts on K(constructible sheaves on $G/B$) by Hecke correpondences and on K(coherent sheaves on $G/B$) by Lusztig's construction [1]. Now we ...
- 3,752
5
votes
2
answers
429
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Parabolic induction, and tensoring (Iwahori/affine) Hecke algebras
In several places, such as [1] and [2], it seems to be implicitly known that (normalized) parabolic induction corresponds to tensoring affine Hecke algebras. I say implicitly because both [1] and [2] ...
5
votes
1
answer
344
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Are parabolic Kazhdan-Lusztig polynomials truncations of the usual Kazhdan-Lusztig polynomials?
Let $(W,S)$ be the affine Weyl group associated to a simple root system.
For $x,y \in W$ we have the usual Kazhdan--Lustig polynomials $h_{y,x} \in \mathbb{Z}[v]$ in Soergel's normalisation, and if ...
- 10.5k
5
votes
0
answers
127
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Is there a derived version of affine Schur-Weyl duality?
One version of affine Schur-Weyl duality states that there is a fully faithful functor from representation of $A_r$ affine Hecke algebra to the representation of $A_n$ affine Lie group assuming $r<...
- 189
5
votes
0
answers
404
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p-adic Hecke operators in the Iwahori-Hecke algebra $C_c(J\backslash G(F)/J)$
$\DeclareMathOperator\ch{ch}$Let $F$ be a non-archimedean local field, $\mathcal{O}$ its ring of integers, $\mathfrak{p}$ its maximal ideal and $\pi$ a uniformizer. I shall use $\kappa(F)$ to denote ...
- 748
5
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0
answers
120
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Towers of algebras, their 2-step centralizer algebras, and analogues of the degenerate affine Hecke algebra
Let $\, \big( {\frak{F}}_0 \subset {\frak{F}}_1 \subset {\frak{F}}_2 \subset \cdots \big)$ be a tower of semi-simple, finite dimensional, unital, complex algebras starting with ${\frak{F}}_0 \cong \...
- 2,137
5
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0
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207
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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 $\{C_w\}$ be the Kazhdan-Lusztig $C$-basis (an answer in terms of the $...
- 413
4
votes
1
answer
250
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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{...
- 71
4
votes
0
answers
189
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Gelfand's transform for noncommutative $C^*$-algebras
Please excuse me if this is well-known, I am not very familiar with the general theory of $C^*$-algebras.
Let $A$ be a unital separable liminal $C^*$-algebra (in the case I am interested in, ...
- 3,446
3
votes
0
answers
72
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How to multiply dots with Young idempotents in the degenerate affine Hecke algebra (type A)
Let $\widehat{\cal H}_n$ be the type A degenerate affine Hecke algebra on $n$ strands, and let $x_1,\cdots,x_n$ be the dots. Inside of this algebra lies the algebra $\mathbb C S_n$, and the Young ...
- 311
2
votes
0
answers
97
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An injective map in equivariant algebraic K-theory
Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$ and $\mathfrak{g}$ be its Lie algebra. Let $\mathcal{N}$ be the nilpotent cone of $\mathfrak{g}$ consists of all nilpotent ...
- 653
1
vote
0
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48
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Length of the product of two elements of the subregular two-sided cell in the affine Weyl group of type A
The affine Weyl group of type $A_n$ can be described as follows. It is the group of all permutations $\sigma: \mathbb Z \to \mathbb Z$ such that $\sigma(i+n)=\sigma(i)+n$ and $\sum_{i=1}^n (\sigma(i)-...
- 2,964
1
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0
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100
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Automorphisms of Iwahori/affine Hecke algebras
Has there been any serious study of automorphisms of extended affine Hecke algebras? Has anyone determined the automorphism group of say, type A extended affine Hecke algebras? I ask because the ...
1
vote
0
answers
36
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When is an affine left cell finite?
Consider an affine Weyl group $\hat W$ of a simple Lie type. Let $w \in \hat W$ and let $C^L(w)$ denote the left cell in $\hat W$ containing $w$.
Is there a good criterion to test whether $C^L(w)$ has ...
- 403
1
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0
answers
66
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Coincidence of notation in the classification of representations of affine Hecke algebras
This is spurred by a short discussion I had in the comments of this MO question.
In Ginzburg's 1998 paper, https://arxiv.org/abs/math/9802004v3, or equivalently in the book by Chriss and Ginzburg, &...
- 159
1
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0
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57
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Block sum for degenerate affine Hecke algebras
The degenerate affine Hecke algebra $H_k$ over a field $F$ is the algebra with generators $s_1,\ldots,s_{k-1}$ and $x_1,\ldots,x_k$, subject to the following relations:
$s_is_js_i=s_js_is_j$ for $i=j\...
- 721
1
vote
0
answers
121
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Coefficient ring of Satake isomorphism
Let $G$ be a split reductive group over local field $F$, $G^L$ be the (complex) Langlands dual group of $G$. Denote $H$ to be the $\mathbb{Z}$-Hecke algebra of $G$, that is the ring of $G(\mathcal{O}...
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