All Questions
Tagged with reductive-groups rt.representation-theory
175 questions
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Linear algebraic group, absolute root system, computing roots
Let $G(F)$ be a reductive linear algebraic group, where $F$ is a local field. Let $T(F)$ be a maximal anisotropic torus of $G$ that splits over a quadratic extension of $F$. Is there an efficient ...
- 11
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87
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Doubling constructions beyond classical groups: general principles?
The doubling method for constructing integral representations of L-functions has been successfully applied to classical groups, as demonstrated in this paper. However, extending this method to a wider ...
- 111
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97
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Centralizer bound for irreducible representations of $\operatorname{SU}_n(\mathbb{C})$
Let $G$ be a finite group, $χ$ be the character of an irreducible representation $V$ of $G$, and $g ∈ G$. Then a classical bound on the trace of $g$ is given by: $|χ(g)|² ≤ |C_G(g)|$, where $C_G(g)$ ...
- 2,907
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44
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Nonvanishing of Jacquet modules of principal series subquotients
Let $G$ be a connected reductive group over a $p$-adic field $F$. For simplicity, we assume $G$ to be split. Fix a Borel $B=TU$ with its Levi decomposition ($U$ unipotent radical, $T$ maximal torus). ...
- 709
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77
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Asymptotic behavior of Shalika germs near non-regular elements
Let $G$ be a connected reductive group over a $p$-adic field $F$. Let $T\subset G$ be a maximal torus. Fix a special maximal compact subgroup $K$ of $G(F)$ and for any closed subgroup $H\subset G(F)$ ...
- 709
3
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1
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170
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Factoring out an element of a root subgroup to make a conjugation integral
Fix a nonarchimedean local field $L/\mathbb{Q}_p$ with ring of integers $\mathcal{O}$, uniformizer $\varpi$, and residue field $k$. If I take a matrix
$$\begin{pmatrix} a & \varpi b \\ c & d \...
- 503
4
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1
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130
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Do parabolic inductions share a composition factor if and only if the inducing data are associate?
Let $F$ be a local field of characteristic zero and $G$ a connected reductive group over $F$. Let us call an inducing datum a triple $(P,M,\sigma)$, where $P$ is a parabolic subgroup of $G$, $M$ is a ...
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4
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415
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Interpretation of the algebra of natural endomorphisms of the fiber functor of $\operatorname{Rep}(G)$
Let $G$ be a connected algebraic group over an algebraically closed field $k$ of characteristic zero (I'm mostly interested in the case of a reductive group).
By the Tannakian formalism, $G(k)$ can be ...
- 3,446
3
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107
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Representations of a reductive Lie group vie central character and K-types
Let $G$ be a real reductive group, let $\widehat G$ denote the unitary dual and $\widehat G_{adm}\supset\widehat G$ be the admissible dual, i.e., the set of isomorphy classes of irreducible Harish-...
- 460
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228
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Chevalley-Solomon formula and Weyl character formula
Let $\Phi\subset V$ be a root system of rank $r$ with Weyl group $W$, a choice of positive roots $\Phi_+$ and exponents $d_1, \ldots, d_r$ (i.e. the invariant algebra $(\operatorname{Sym}^\bullet V)^W$...
- 3,446
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266
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References for $K$-orbits in $G/B$
Let $G$ be a reductive group, $K$ a symmetric subgroup of $G$ (e.g., fixed point of an involution), and $B$ a Borel subgroup of $G$. Then it is well known that $G/B$ has finitely many $K$-orbits. ...
- 741
13
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509
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Is there a simple proof that representations of GL(n,k) are determined by their restriction to diagonal matrices?
Let $k$ be a field of characteristic zero. The general linear group $\mathrm{GL}(n,k)$ has a subgroup $\mathrm{D}(n,k)$ consisting of invertible diagonal matrices. These are linear algebraic groups ...
- 22.3k
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76
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Simplest way to classify reducibility of principal series for $p$-adic $\mathrm{SL}_2$
Let $F$ be a $p$-adic field and $G_1=\mathrm{SL}_2(F)\subset \mathrm{GL}_2(F)=G$. For simplicity, we assume $p>2$. Denote by $|\cdot|$ the normalized absolute value on $F$. Here I shall focus on ...
- 709
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213
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Gelfand-Kazhdan criterion, exposition by Paul Garrett
Here is Paul Garrett's exposition on the Gelfand-Kazhdan Criterion.
In page 4 of the exposition, he showed the following lemma.
Lemma (Page 4): Let $B, t, S$ be as above and for $\alpha, \beta$ in $...
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1
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203
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Orbital integrals of $\operatorname{SL}_2$ and the fundamental lemma
$\DeclareMathOperator\SL{SL}$When I was checking some orbital integral computations of Sally-Shalika's The Plancherel Formula for $\SL_2$ over a Local Field, Proceedings of the National Academy of ...
- 709
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110
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On the character of a representation of $\mathrm{GL}(n,\mathbb{R})$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Ad{Ad}$Let $G=\GL(n,\mathbb{R})$. Given a continuous admissible irreducible representation of $G$ in a Frechet (or a Banach) space. Then its character ...
- 21.8k
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73
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Embeddings of unitary groups over $\mathbb{Q}$
$\DeclareMathOperator\GU{GU}$$\DeclareMathOperator\GL{GL}$I'm a bit confused by the following situation:
suppose we have an Hermitian vector space $V=K^3$ of matrix $$
J=\begin{pmatrix}& & \...
- 91
3
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1
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327
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Pro-unipotent radical (in Bruhat-Tits) vs. unipotent radical (and reference request)
I'm a beginner in Bruhat-Tits theory, and the following phenomenon makes me puzzled.
So let $F$ be a $p$-adic field, with $\mathfrak{o}\supset \mathfrak{p}$ its ring of integers and maximal ideal, and ...
- 709
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2
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314
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Holomorphic discrete series vs. discrete series
(I apologize in advance if this question is too naive for experts.) Let $G$ be a real semisimple Lie group. I know that holomorphic discrete series representations are only a part of all the discrete ...
- 709
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50
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Do parabolic/Levi pairs admit dynamic descriptions over disconnected base?
In Gille, Thm. 7.3.1, it is proven that given a reductive group scheme $G \to S$ over a connected base $S$, every parabolic-Levi pair $(P, L)$ over $S$ admits a dynamic description, i.e. is of the ...
- 605
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268
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Stabilizer of a Levi subgroup in the Weyl group and its quotient
(I appologize in advance if this question is too naive for experts, since I know very little about the geometry/combinatorics of Weyl/Coxeter groups.)
For simplicity, let $G$ be a connected reductive ...
- 709
4
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168
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Representation rings of disconnected reductive groups
Let $G$ be a disconnected complex reductive group, or equivalently a disconnected compact real Lie group, the kind treated by Segal in his (first ever, possibly) paper "The representation-ring of ...
- 413
4
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1
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185
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Canonicality of group of integers for reductive groups over non-Archimedean local field
$\DeclareMathOperator\GL{GL}$Let $G$ be a semisimple (but I think there is no obstruction to assume it to be reductive) algebraic group over a non-Archimedean local field $K$ and $\mathcal{O}_K$ be ...
- 6,028
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134
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Tempered representations and unramified principal series
For $V$ a tempered representation of connected reductive group over a local field of characteristic zero. I want to show that for an Iwahori subgroup $B$, the set of fixed points $V^B\neq 0$, thereby ...
2
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1
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92
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Image of the intertwining operator for GL(2) is $K$-invariant at the "pole" $s=1$
I am taking a look at the residues of Eisenstein series and have a question about a local computation. Let $k$ be a local field, $G = \operatorname{GL}_2(k)$, and $P$ (resp. $K$) the standard ...
- 6,180
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430
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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] ...
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1
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112
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Continuity of central character [closed]
Let $G$ be a $p$-adic reductive group and $Z\subseteq G$ the center. Let $\pi$ be an irreducible admissible representation of $G$. By Schur's lemma, it is easy to show that there is a group ...
- 833
3
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1
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190
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Quantum group associated to a reductive group
In most of the classical references about quantum groups, these objects are defined as a one-parameter deformation of the universal enveloping algebra. However, I have read in several papers that it ...
- 31
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297
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Orbit of a parahoric subgroup on a flag variety
Let $G$ be a split reductive group over a nonarchimedean local field $F$ (I'm particularly interested in the case of $\operatorname{GSp}_{2n}$).
Given a parahoric subgroup $K \subset G(F)$, and a ...
- 37k
4
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299
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Can any pair of associate parabolics be related by opposite parabolics?
Let $G$ be a reductive group, say over an algebraically closed field of characteristic zero.
We have the following definitions for a pair of parabolic subgroups $P_1$ and $P_2$ with Levi quotients $...
4
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2
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231
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Weight of the defining function of a Bruhat cell in a simply connected semisimple group
Let $G$ be a connected semisimple simply connected group over an algebraically connected field. Let $w_0$ be the longest element in the Weyl group $W$ and $s_i$ be a simple reflection in $W$. Choose a ...
- 291
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77
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Conjugacy of cocharacters from conjugacy of labelled diagrams
Everything to follow is over some fixed algebraically closed field $k$. Although all the definitions make sense regardless of characteristic, the meat of the question is about small positive ...
- 12.9k
3
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1
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181
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Does $\mathrm{Ext}^i_G(\pi,\pi')$ vanish if $\pi$ and $\pi'$ are smooth irreducible representations of $G$ with different central characters?
Let $G$ be a $p$-adic group, ie. the group of $F$-points of some connected reductive group over $F$, where $F$ is a $p$-adic field. We consider complex smooth representations of $G$. Any irreducible ...
- 769
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175
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Centralizer of a reductive subgroup
Let $G$ be a reductive group over $\mathbb{C}$ and $H\subseteq G$ a reductive subgroup. Let $\rho$ be a faithful irreducible finite dimensional representation of $G$ over $\mathbb{C}$. Assume that $\...
- 833
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206
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Springer sheaf and Deligne-Lusztig induction
Let $G=Gl_n$ be the general linear group over the algebraic closure of a finite field $\overline{\mathbb{F}}_q $ and let $F:G \to G$ be the standard Frobenius. On $G$ there is the Springer (perverse) ...
- 1,061
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71
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Principal series representations for complex groups
Let $G$ be a complex semisimple group.
In Bernstein-Gelfand, "Tensor products of finite and infinite dimensional representations of semisimple Lie algebras" (http://www.numdam.org/article/...
- 21
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177
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How do characters of representations in cohomology depend on the (positive-characteristic) field?
The following sentence appears in Jantzen - Representations of algebraic groups, 2nd edition, p. x, where $G$ is a reductive group over an algebraically closed field $k$, $B$ is a Borel subgroup, $T$ ...
- 12.9k
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101
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Number of points of parabolic Springer fibres for general reductive groups
My question is the same as this post but for an arbitrary reductive $G$ instead of just $\mathrm{GL}_n$.
Let $G$ be a connected split reductive group over a finite field $k$.
Let $P$ be a parabolic ...
- 2,751
4
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108
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When does the null-cone consist entirely of eigenvectors?
Let $V$ be a finite-dimensional representation of a complex reductive Lie algebra $\mathfrak g$.
For our purposes, we may define the null-cone like this: $v\in V$ belongs to the null-cone if and only ...
- 18.4k
3
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203
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A quantity computed from weights of representations -- Have you seen it?
The following quantity has come up in some work my collaborators and I are doing on equivariant D-modules, and in that particular context it seems to be very significant (i.e. it's the only "...
- 3,079
2
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1
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148
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Does the F-unitary group isomorphism arises from a conformal isometry?
Let $K$ be a CM-field with totally real subfield $F$. Let $(V_1, h_1)$ and $(V_2, h_2)$ be two $n$-dimensional $K$-vector spaces with nondegenerate Hermitian forms, where $n\geq 3$.
Question 1 Does ...
- 337
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147
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Automorphism groups of "reductive" Lie algebras in positive characteristic
I put "reductive" in quotes because, of course, in positive characteristic one should speak of Lie algebras of reductive groups, not of reductive Lie algebras.
Let $G$ be a reductive group ...
- 12.9k
5
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279
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Asymptotic behavior of matrix coefficients
I'm reading Casselman's notes "Introduction to the theory of admissible representations of p-adic reductive groups". In chapter 4 "The asymptotic behavior of matrix coefficients", ...
- 121
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792
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Constructing a Kac-Moody group as a quotient of the free product of its root subgroups
The paper "Regular Functions on Certain Infinite-dimensional Groups" by Kac and Peterson describes the construction of a group associated to the datum of a Kac-Moody algebra in a way I haven'...
- 53
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271
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Can closure of an orbit under a reductive action contain infinitely many orbits?
I posted this on math.se a week ago, currently it has 23 views and no other feedback.
Here on MO there are several questions about orbit closures but I could not find anything about what I need.
To be ...
- 18.4k
2
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156
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Semisimple covers of varieties
Let $X$ be an algebraic variety.
The finite étale covers of $X$ are measured by the étale fundamental group $\pi_1^{\rm et}(X)$.
On the other hand, the Cox ring ${\rm Cox}(X)$ of $X$ (finitely ...
- 1,585
12
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1
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569
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Reference for character sheaves over $\mathrm{GL}_n(q)$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}$I know a little bit about complex representation theory of finite reductive groups as $\GL_n(q),\SO_n(q)$ etc via Deligne-Lusztig induction and ...
- 1,061
8
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0
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267
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A Lie-theoretic question regarding $B\ltimes \mathfrak{g}/\mathfrak{b}$
I am stuck on a seeming elementary Lie-theoretic question arising from a study of components of affine Springer fibers. Will be very grateful if somebody would like to share some insight, or ...
- 1,856
4
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1
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169
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Divisibility by 2 of invariants forms on reductive Lie algebras and anomaly cancellation for gauge theories
Let $G$ be a connected reductive group over $\mathbb C$ and let $\rho:G\to \operatorname{Sp}(2n,\mathbb C)$ be a homomorphism. You can think about $\rho$ as a linear symplectic representation of $G$ ...
- 7,037
3
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0
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79
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Gindikin-Karpelevich formula in the quasi-split case
In Euler Products (1971), Langlands established the Gindikin-Karpelevich formula for split reductive groups over the local fields $\mathbb{Q}_p$. On the other hand, the archimedean Gindikin-...
- 193