Centers and Cocenters of $0$-Hecke algebras

关于类型$G(r,1,n)$的循环Hecke代数的余核

On the cocenter of the cyclotomic Hecke algebra of type $G(r,1,n)$

本文基于Geck和Pfeiffer的工作，构建了类型$G(r,1,n)$的循环Hecke代数$\mathscr{H}_{n,K}$的整数基，并证明其余核和中心的维度与基域特征、Hecke参数和循环参数无关。

分圆Hecke代数上的迹型与分圆Schur代数的余中心

Trace forms on the cyclotomic Hecke algebras and cocenters of the cyclotomic Schur algebras

本文在A型分圆Hecke代数上定义了统一的迹型，并构造了双基及分圆Schur代数余中心的显式基，证明其维数与基域、Hecke参数及分圆参数无关。

Cocenters of $p$-adic groups, III: Elliptic and rigid cocenters

In this paper, we show that the elliptic cocenter of the Hecke algebra of a connected reductive group over a nonarchimedean local field is contained in the rigid cocenter. As applications, we prove the trace Paley-Wiener theorem and the abstract Selberg principle for mod-$l$ representations.

Jordan Decompositions of cocenters of reductive $p$-adic groups

Cocenters of Hecke algebras $\mathcal H$ play an important role in studying mod $\ell$ or $\mathbb C$ harmonic analysis on connected $p$-adic reductive groups. On the other hand, the depth $r$ Hecke algebra $\mathcal H_{r^+}$ is well suited to study depth $r$ smooth representations. In this paper, we study depth $r$ rigid cocenters $\overline{\mathcal H}^{\mathrm{rig}}_{r^+}$ of a connected reductive $p$-adic group over rings of characteristic zero or $\ell\neq p$. More precisely, under some mild hypotheses, we establish a Jordan decomposition of the depth $r$ rigid cocenter, hence find an exp…

Cocenter of $p$-adic groups, II: induction map

In this paper, we study some relation between the cocenter $\bar H(G)$ of the Hecke algebra $H(G)$ of a connected reductive group $G$ over an nonarchimedean local field and the cocenter $\bar H(M)$ of its Levi subgroups $M$. Given any Newton component of $\bar H(G)$, we construct the induction map $\bar i$ from the corresponding Newton component of $\bar H(M)$ to it. We show that this map is surjective. This leads to the Bernstein-Lusztig type presentation of the cocenter $\bar H(G)$, which generalizes the work \cite{HN2} on the affine Hecke algebras. We also show that the map $\bar i$ we cons…

The cocenter-representation duality

Affine Hecke algebras arise naturally in the study of smooth representations of reductive $p$-adic groups. Finite dimensional complex representations of affine Hecke algebras (under some restriction on the isogeny class and the parameter function) has been studied by many mathematicians, including Kazhdan-Lusztig \cite{KL}, Ginzburg \cite{CG}, Lusztig \cite{L1}, Reeder \cite{Re}, Opdam-Solleveld \cite{OS}, Kato \cite{Kat}, etc. The approaches are either geometric or analytic. In this note, we'll discuss a different route, via the so-called "cocenter-representation duality", to study finite dim…

$P$-alcoves, parabolic subalgebras and cocenters of affine Hecke algebras

This is a continuation of the sequence of papers \cite{HN2}, \cite{H99} in the study of the cocenters and class polynomials of affine Hecke algebras $\ch$ and their relation to affine Deligne-Lusztig varieties. Let $w$ be a $P$-alcove element, as introduced in \cite{GHKR} and \cite{GHN}. In this paper, we study the image of $T_w$ in the cocenter of $\ch$. In the process, we obtain a Bernstein presentation of the cocenter of $\ch$. We also obtain a comparison theorem among the class polynomials of $\ch$ and of its parabolic subalgebras, which is analogous to the Hodge-Newton decomposition theor…

Cocenters of $p$-adic groups, I: Newton decomposition

In this paper, we introduce the Newton decomposition on a connected reductive $p$-adic group $G$. Based on it we give a nice decomposition of the cocenter of its Hecke algebra. Here we consider both the ordinary cocenter associated to the usual conjugation action on $G$ and the twisted cocenter arising from the theory of twisted endoscopy. We give Iwahori-Matsumoto type generators on the Newton components of the cocenter. Based on it, we prove a generalization of Howe's conjecture on the restriction of (both ordinary and twisted) invariant distributions. Finally we give an explicit description…

Kac-Moody 群的 Hecke 代数的余中心

Cocenter of Hecke algebras of Kac-Moody groups

本文证明了 Kac-Moody 群泛型 Hecke 代数的余中心由共轭类典范元素构成基，通过重整化轨道积分新框架解决了线性无关性开放问题。

仿射渐近Hecke代数的结构

On the structure of the affine asymptotic Hecke algebras

本文通过构造非平凡中心扩张的例子证明Lusztig猜想的较弱形式是最优的，同时证明仿射Hecke代数到仿射渐近Hecke代数的Lusztig同态诱导上中心同构，并探讨中心扩张与上中心结构的关联。

Braverman-Kazhdan 渐近 Hecke 代数的迹 Paley-Wiener 定理

Trace Paley-Wiener theorem for Braverman-Kazhdan's asymptotic Hecke algebra

本文研究 Braverman-Kazhdan 渐近 Hecke 代数，通过分类其不可约表示证明余中心同构猜想，并在 GL_n 情形给出显式描述。

Centers, cocenters and simple quantum groups

We define the notion of a (linearly reductive) center for a linearly reductive quantum group, and show that the quotient of a such a quantum group by its center is simple whenever its fusion semiring is free in the sense of Banica and Vergnioux. We also prove that the same is true of free products of quantum groups under very mild non-degeneracy conditions. Several natural families of compact quantum groups, some with non-commutative fusion semirings and hence very "far from classical", are thus seen to be simple. Examples include quotients of free unitary groups by their centers, recovering p…

任意Coxeter群Hecke代数中的中心化子

Centralizers in Hecke Algebras of Any Coxeter Group

针对任意Coxeter群Hecke代数中抛物子代数中心化子在不定情形下研究不足的问题，结合代数技术与Davis复形几何工具，给出中心化子的基、分类无限Coxeter群有限部分共轭类并定义适配的类多项式变体，推广了中心相关已知结果。

On the center conjecture for the cyclotomic KLR algebras

The center conjecture for the cyclotomic KLR algebras $R_β^Λ$ asserts that the center of $R_β^Λ$ consists of symmetric elements in its KLR $x$ and $e(ν)$ generators. In this paper we show that this conjecture is equivalent to the injectivity of some natural map $\barι_β^{Λ,i}$ from the cocenter of $R_β^Λ$ to the cocenter of $R_β^{Λ+Λ_i}$ for all $i\in I$ and $Λ\in P^+$. We prove that the map $\barι_β^{Λ,i}$ is given by multiplication with a center element $z(i,β)\in R_β^{Λ+Λ_i}$ and we explicitly calculate the element $z(i,β)$ in terms of the KLR $x$ and $e(ν)$ generators. We present an explic…

On the center of quiver-Hecke algebras

We compute the equivariant cohomology ring of the moduli space of framed instantons over the affine plane. It is a Rees algebra associated with the center of cyclotomic degenerate affine Hecke algebras of type A. We also give some related results on the center of quiver Hecke algebras and cohomology of quiver varieties.

渐近仿射Hecke代数的一个几何实现

A geometric realization of the asymptotic affine Hecke algebra

本研究证明了渐近仿射Hecke代数$J$的几何描述猜想，给出其不可约表示参数化的新几何证明、谱描述的新证明及余中心描述的证明，基于对$J$的新代数描述。

Centers of degenerate cyclotomic Hecke algebras and parabolic category O

We prove that the center of each degenerate cyclotomic Hecke algebra associated to the complex reflection group of type B_d(l) consists of symmetric polynomials in its commuting generators. The classification of the blocks of the degenerate cyclotomic Hecke algebras is an easy consequence. We then deduce that the center of an integral block of parabolic category O for the Lie algebra gl_n(C) is generated by the center of its universal enveloping algebra.

Modified affine Hecke algebras and quiver Hecke algebras of type $A$

We introduce some modified forms for the degenerate and non-degenerate affine Hecke algebras of type $A$. These are certain subalgebras living inside the inverse limit of cyclotomic Hecke algebras. We construct faithful representations and standard bases for these algebras and give some explicit description of their centers. We show that there are algebra isomorphisms between some generalized Ore localizations of these modified affine Hecke algebras and of the quiver Hecke algebras of type $A$. As an application, we show that the center conjecture for the cyclotomic quiver Hecke algebra of typ…

偏微分Hecke代数、Schur-Weyl对偶性与Frobenius特征公式

Mirabolic Hecke algebras, Schur-Weyl duality and Frobenius character formulas

本文研究了偏微分Hecke代数的新的呈现方式，建立了其与量子群之间的Schur-Weyl对偶性，并推导出Frobenius特征公式及递归的Murnaghan-Nakayama规则。

On the centers of cyclotomic quiver Hecke algebras

Let $n\in\mathbb{N}$ and $K$ be any field. For any symmetric generalized Cartan matrix $A$, any $β$ in the positive root lattice with height $n$ and any integral dominant weight $Λ$, one can associate a quiver Hecke algebras $R_β(K)$ and its cyclotomic quotient $R_β^Λ(K)$ over $K$. It has been conjectured that the natural map from $R_β(K)$ to $R_β^Λ(K)$ maps the center of $R_β(K)$ surjectively onto the center of $R_β^Λ(K)$. A similar conjecture claims that the center of the affine Hecke algebra of type $A$ maps surjectively onto the center of its cyclotomic quotient---the cyclotomic Hecke alge…

类型的 Hecke 代数的中心

The center of Hecke algebras of types

本文研究附属于 Bernstein 块的类型的 Hecke 代数中心，并在驯分裂约化群等条件下描述相关 Bernstein 中心。

nil-Hecke 代数的根幂商与 Weyl 不变式理论

Root-power quotients of nil-Hecke algebras and Weyl invariant theory

本文研究由根坐标幂定义的 nil-Hecke 商，通过根轨道幂理想和判别式消去确定中心，计算 Hilbert 级数并分类自内射性，实现为 Cherednik 商。研究问题为根幂商的结构，核心方法是根轨道理想与相对判别式，主要贡献是完整分类与实现。

Buildings and Hecke algebras

This paper investigates the connections between buildings and Hecke algebras through the combinatorial study of two algebras spanned by averaging operators on buildings. As a consequence we obtain a geometric and combinatorial description of certain Hecke algebras, and in particular of the Macdonald spherical functions and the center of affine Hecke algebras. The results of this paper are used in later work to study spherical harmonic analysis on affine buildings, and to study isotropic random walks on affine buildings.

Finiteness for Hecke algebras of $p$-adic groups

Let $G$ be a reductive group over a non-archimedean local field $F$ of residue characteristic $p$. We prove that the Hecke algebras of $G(F)$ with coefficients in a ${\mathbb Z}_{\ell}$-algebra $R$ for $\ell$ not equal to $p$ are finitely generated modules over their centers, and that these centers are finitely generated $R$-algebras. Following Bernstein's original strategy, we then deduce that "second adjointness" holds for smooth representations of $G(F)$ with coefficients in any ring $R$ in which $p$ is invertible. These results had been conjectured for a long time. The crucial new tool tha…

The centers of Iwahori-Hecke algebras are filtered

We show that the center of the Iwahori--Hecke algebra of the symmetric group $S_n$ carries a natural filtered algebra structure, and that the structure constants of the associated graded algebra are independent of $n$. A series of conjectures and open problems are also included.

Some multiplication formulas in an affine Hecke algebra

In this paper we consider the Hecke algebra $\mathcal {H}$ associated to an extended affine Weyl group of type $\widetilde{B_2}$. We give some interesting formulas on $C_{rt}S_λ$, which imply some relations between the Kazhdan-Lusztig coefficients $μ(y,w)$ and representations of some algebraic groups. Here $C_{rt}$ is one element in the Kazhdan-Lusztig basis of $\mathcal {H}$ and $S_λ$ is an element in the center of $\mathcal {H}$.

Center and representations of infinitesimal Hecke algebras of sl_2

In this paper, we compute the center of the infinitesimal Hecke algebras Hz associated to sl_2 ; then using nontriviality of the center, we study representations of these algebras in the framework of the BGG category O. We also discuss central elements in infinitesimal Hecke algebras over gl(n) and sp(2n) for all n. We end by proving an analogue of the theorem of Duflo for Hz.

Centers of Hecke Algebras of Complex Reflection Groups

We provide a dual version of the Geck--Rouquier Theorem on the center of an Iwahori--Hecke algebra, which also covers the complex case. For the eight complex reflection groups of rank $2$, for which the symmetrising trace conjecture is known to be true, we provide a new faithful matrix model for their Hecke algebra $H$. These models enable concrete calculations inside $H$. For each of the eight groups, we compute an explicit integral basis of the center of $H$.

On the spectral decomposition of affine Hecke algebras

An affine Hecke algebra H contains a large abelian subalgebra A. The center Z of H is the subalgebra of Weyl group invariant elements in A. The natural trace of the affine Hecke algebra can be written as an integral of a rational $n$ form (with values in the linear dual of H) over a certain cycle in the algebraic torus T=spec(A). We derive the Plancherel formula of the affine Hecke algebra by localization of this integral on a certain subset of spec(Z).

Products of Geck-Rouquier conjugacy classes and the Hecke algebra of composed permutations

We show the q-analog of a well-known result of Farahat and Higman: in the center of the Iwahori-Hecke algebra $H_{n,q}$, if $(a_{λμ}^ν(n,q))_ν$ is the set of structure constants involved in the product of two Geck-Rouquier conjugacy classes $Γ_{λ,n}$ and $Γ_{μ,n}$, then each coefficient $a_{λμ}^ν(n,q)$ depends on $n$ and $q$ in a polynomial way. Our proof relies on the construction of a projective limit of the Hecke algebras; this projective limit is inspired by the Ivanov-Kerov algebra of partial permutations.

Combinatorics of Centers of 0-Hecke Algebras in Type A

A basis of the center of the 0-Hecke algebra of an arbitrary finite Coxeter group was described by He in 2015. This basis is indexed by certain equivalence classes of the Coxeter group whose explicit description is rather complicated. Even their number is not obvious. We consider case of the symmetric group $\mathfrak{S}_n$. Building on work of Geck, Kim and Pfeiffer, we obtain a complete set of representatives of the equivalence classes. This set is naturally parametrized by certain compositions of $n$ called maximal. It follows that the maximal compositions of $n$ index the basis of the cent…

The mirabolic Hecke algebra

The Iwahori-Hecke algebra of the symmetric group is the convolution algebra of $\gl_n$-invariant functions on the variety of pairs of complete flags over a finite field. Considering convolution on the space of triples of two flags and a vector we obtain the mirabolic Hecke algebra $R_n$, which had originally been described by Solomon. In this paper we give a new presentation for $R_n$ which shows that it is a quotient of a cyclotomic Hecke algebra, as defined by Ariki and Koike. From this we recover the results of Siegel about the representations of $R_n$. We use Jucys-Murphy elements to descr…

Generic pro-$p$ Hecke algebras

This is an extended and corrected version of the author's Diplomarbeit. A class of algebras called generic pro-$p$ Hecke algebras is introduced, enlarging the class of generic Hecke algebras by considering certain extensions of (extended) Coxeter groups. Examples of generic pro-$p$ Hecke algebras are given by pro-$p$-Iwahori Hecke algebras and Yokonuma-Hecke algebras. The notion of an orientation of a Coxeter group is introduced and used to define `Bernstein maps' intimately related to Bernstein's presentation and to Cherednik's cocycle. It is shown that certain relations in the Hecke algebra …

Base Change for the Iwahori-Hecke Algebra of GL_2

Let $F$ be a non-Archimedean local field of characteristic not equal to 2, let $E/F$ be a finite unramified extension field, and let $σ$ be a generator of $\text{Gal}(E, F )$. Let $f$ be an element of $Z(H_{I_E})$, the center of the Iwahori-Hecke algebra for $GL_2(E)$, and let $b$ be the Iwahori base change homomorphism from $Z(H_{I_E})$ to $Z(H_{I_F})$, the center of the Iwahori-Hecke algebra for $GL_2(F)$ [8]. This paper proves the matching of the $σ$-twisted orbital integral over $GL_2(E)$ of $f$ with the orbital integral over $GL_2(F)$ of $bf$. To do so, we compute the orbital and $σ$-twis…

Infinitesimal Hecke algebra of sl_2 in positive characteristic

In this paper we consider an infinitesimal Hecke algebra of $sl_2$ in positive characteristic. We show that it is a finitely generated module over its center, and the smooth and the Azumaya loci of its center coincide.

局部域上 Kac-Moody 群的完备 Iwahori-Hecke 代数与 parahoric Hecke 代数

Completed Iwahori-Hecke algebras and parahoric Hecke algebras for Kac-Moody groups over local fields

本文定义了局部域上 split Kac-Moody 群的 Iwahori-Hecke 代数完备化，利用 masure 工具确定其中心并证明同构于球面 Hecke 代数，同时将 Hecke 代数构造推广至特殊球面面。

关于一般仿射Hecke代数的中心

On the center of the generic affine Hecke algebra

本文确定了一般仿射Hecke代数的中心为优势腔半群代数，并通过极大交换子代数将其与环面簇的量子上同调联系起来。

$G(r,p,n)$ 型旋轮 Hecke 代数的半正规基

Seminormal basis for the cyclotomic Hecke algebra of type $G(r,p,n)$

本文研究了$G(r,p,n)$型旋轮Hecke代数的半正规基，通过证明$γ$-系数的有理性质，利用$H_{r,n}$的半正规基构造了$H_{r,p,n}$的半正规基，并得到了中心及$σ$-扭曲$k$-中心的显式基。

A tale of two Hecke algebras

We use Bernstein's presentation of the Iwahori-Matsumoto Hecke algebra to obtain a simple proof of the Satake isomorphism and, in the same stroke, compute the center of the Iwahori-Matsumoto Hecke algebra.

Frobenius map for the centers of Hecke algebras

We introduce a commutative associative graded algebra structure on the direct sum Z of the centers of the Hecke algebras associated to the symmetric groups in n letters for all n. As a natural deformation of the classical construction of Frobenius, we establish an algebra isomorphism from the algebra Z to the ring of symmetric functions. This isomorphism provides an identification between several distinguished bases for the centers (introduced by Geck-Rouquier, Jones, Lascoux) and explicit bases of symmetric functions.

Local Langlands Correspondence for Classical Groups and Affine Hecke Algebras

Using the results of J. Arthur on the representation theory of classical groups with additional work by Colette Moeglin and its relation with representations of affine Hecke algebras established by the author, we show that the category of smooth complex representations of a split $p$-adic classical group and its pure inner forms is naturally decomposed into subcategories which are equivalent to a tensor product of categories of unipotent representations of classical groups (in the sense of G. Lusztig). A statement of this kind had been conjecture by G. Lusztig. All classical groups (general li…

局部域上 Kac–Moody 群的完备化 Iwahori–Hecke 代数

Completed Iwahori-Hecke algebra for Kac-Moody groups over local fields

本文针对局部域上分裂 Kac–Moody 群，利用 Weyl 几乎有限支撑的 Iwahori 双不变函数构造完备化 Iwahori–Hecke 代数，并证明其中心含球面 Hecke 代数。

Hecke-Clifford algebras and spin Hecke algebras I: the classical affine type

Associated to the classical Weyl groups, we introduce the notion of degenerate spin affine Hecke algebras and affine Hecke-Clifford algebras. For these algebras, we establish the PBW properties, formulate the intertwiners, and describe the centers. We further develop connections of these algebras with the usual degenerate (i.e. graded) affine Hecke algebras of Lusztig by introducing a notion of degenerate covering affine Hecke algebras.

Matching of orbital integrals (transfer) and Roche Hecke algebra isomorphisms

Let $F$ be a non-Archimedan local field, $G$ a connected reductive group defined and split over $F$, and $T$ a maximal $F$-split torus in $G$. Let $χ_0$ be a depth zero character of the maximal compact subgroup $\mathcal{T}$ of $T(F)$. It gives by inflation a character $ρ$ of an Iwahori subgroup $\mathcal{I}$ of $G(F)$ containing $\mathcal{T}$. From Roche, $χ_0$ defines a split endoscopic group $G'$ of $G$, and there is an injective morphism of ${\Bbb C}$-algebras $\mathcal{H}(G(F),ρ) \rightarrow \mathcal{H}(G'(F),1_{\mathcal{I}'})$ where $\mathcal{H}(G(F),ρ)$ is the Hecke algebra of compactly…

Gieseker 簇的 K-理论与 A 型循环 Hecke 代数

K-theory of Gieseker variety and type A cyclotomic Hecke algebra

本文通过将 Gieseker 空间的等变 K-理论等同于循环 Hecke 代数的 Jucys-Murphy 中心，给出了 Gieseker 簇的等变 K-理论的代数描述，并讨论了其对循环 Hecke 代数中心及 q=1 和单位根特化的影响。

The Bernstein presentation for general connected reductive groups

Let F be a non-Archimedean local field and let G be a connected reductive affine algebraic F-group. Let I be an Iwahori subgroup of G(F) and denote by H(G; I) the Iwahori-Hecke algebra, i.e. the convolution algebra of complex-valued functions on G(F) which are left- and right-invariant by I-translations. This article proves that the Iwahori-Hecke algebra H(G; I) has both an Iwahori-Matsumoto Presentation and a Bernstein Presentation analogous to those for affine Hecke algebras on root data found in Lusztig's "Affine Hecke algebras and their graded version", and gives a basis (in other words, a…

渐近Hecke范畴的中心与单位特征sheaves

The center of the asymptotic Hecke category and unipotent character sheaves

本文证明了渐近Hecke范畴的中心为模ularity张量范畴，并研究了其在二面体群及H3、H4部分细胞中的应用。

Conjugacy classes of non-translations in affine Weyl groups and applications to Hecke algebras

Let W be an Iwahori-Weyl group of a connected reductive group G over a non-archimedean local field. I prove that if w is an element of W that does not act on the corresponding apartment of G by a translation then one can apply to w a sequence of conjugations by simple reflections, each of which is length-preserving, resulting in an element w' for which there exists a simple reflection s such that l(sw's)>l(w'). Even for affine Weyl groups, a special case of Iwahori-Weyl groups and also an important subclass of Coxeter groups, this is a new fact about conjugacy classes. Further, there are impli…

Iwahori-Hecke Algebras of SL_2 over 2-dimensional Local Fields

In this paper we construct an analogue of Iwahori-Hecke algebras of SL_2 over 2-dimensional local fields. After considering coset decompositions of double cosets of a Iwahori subgroup, we define a convolution product on the space of certain functions on SL_2, and prove that the product is well-defined. Then we investigate the structure of the Hecke algebra. We determine the center of the Hecke algebra and consider Iwahori-Matsumoto type relations.

仿射Hecke代数的刚性对偶复形

Rigid dualizing complexes of affine Hecke algebras

本文确定了约化根系关联的仿射Hecke代数$H_q$的刚性对偶复形，并推导出经典Hecke代数在特定条件下在其中心上为Frobenius的，且Nakayama自同构由显式对合给出。

A tale of parahoric--Hecke algebras, Bernstein and Satake homomorphisms

Let $\mathbf{G}$ be a connected reductive group over a {non-archimedean local field} $F$. Let $K_\mathcal{F}$ be the parahoric subgroup attached to a facet $\mathcal{F}$ in the Bruhat--Tits building of $\mathbf{G}$. The ultimate goal of the present paper is to describe the center of the parahoric--Hecke algebra $\mathcal{H}(\mathbf{G}(F)//K_{\mathcal{F}}, \mathbb{Z}[q^{-1}])$ with level $K_{\mathcal{F}}$ and prove the compatibility of generalized (twisted) Bernstein and Satake homomorphisms.

作为范畴中心的约化群等变导出范畴

Equivariant derived category of a reductive group as a categorical center

本文证明约化群的伴随等变导出范畴等价于环面等变 Hecke 范畴的 monoidal 中心，并据此重新证明 Drinfeld 中心与幂幺特征层范畴的等价性及强中心层刻画。

The base change fundamental lemma for central elements in parahoric Hecke algebras

Clozel and Labesse proved the base change fundamental lemma for spherical Hecke algebras attached to an unramified group over a p-adic field. This paper proves an analogous fundamental lemma for centers of parahoric Hecke algebras attached to the same class of groups. This provides an ingredient needed for the author's program to study Shimura varieties with parahoric level structure at p.

A Hecke Algebra on the Double Cover of a Chevalley Group Over $\mathbb{Q}_{2}$

We prove that a certain genuine Hecke algebra $\mathcal{H}$ on the non-linear double cover of a simple, simply-laced, simply-connected, Chevalley group $G$ over $\mathbb{Q}_{2}$ admits a Bernstein presentation. This presentation has two consequences. First, the Bernstein component containing the genuine unramified principal series is equivalent to $\mathcal{H}$-mod. Second, $\mathcal{H}$ is isomorphic to the Iwahori-Hecke algebra of the linear group $G/Z_{2}$, where $Z_{2}$ is the $2$-torsion of the center of $G$. This isomorphism of Hecke algebras provides a correspondence between certain gen…