# Questions tagged [reductive-groups]

A reductive group is an algebraic group $G$ over an algebraically closed field such that the unipotent radical of $G$ is trivial

314
questions

**3**

votes

**1**answer

90 views

### Action of split torus on positive root spaces

Let $G$ be a connected reductive group over a field $k$ (not necessarily algebraically closed). Let $S$ be a maximal split torus in $G$ with relative root system $\Phi = \Phi_k(S,G)$. Let $\Phi^+$ ...

**3**

votes

**0**answers

73 views

### How to read the paper of Arthur on trace formula on general reductive groups (Reference Request)

My question is about the correct order to read the papers by Arthur on trace formula. Arthur's papers are perfectly well-written, but maybe a little too hard for me to go through easily.
I would like ...

**3**

votes

**2**answers

194 views

### Reductive group with simply connected derived group has all root groups $\mathrm{SL}_2$

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}$Motivation: I am trying to understand why the Deligne-Langlands conjectures are only stated for $p$-adic reductive groups with connected ...

**3**

votes

**0**answers

28 views

### Generating $K$-types of a $(\mathfrak g,K)$-module for $K$ disconnected

Let $G$ be a real reductive Lie group, let $K$ be a maximal compact subgroup of $G$, and let $V$ be a $(\mathfrak g,K)$-module. For $\sigma\in\widehat{K}$ we denote the $\sigma$-isotypic component of $...

**4**

votes

**0**answers

82 views

### The coherence property of center of universal enveloping algebra for reductive Lie algebra?

Let $G' \subset G$ be two reductive Lie groups over $\mathbb{R}$ and $\mathfrak{g}_{\mathbb{C}}' \subset \mathfrak{g}_{\mathbb{C}}$ be their complexified reductive Lie algebra over $\mathbb{C}$, ...

**5**

votes

**1**answer

132 views

### Signs in Chevalley systems for reductive groups

Let $G$ be a pinned split reductive group. There exists a Chevalley system:
For each root $b$ in its root system there are parametrisations $x_b: \mathbb{G}_a \rightarrow U_b$ of the corresponding ...

**3**

votes

**0**answers

53 views

### compactly induction of smooth modules over Hecke algebras

Let $G$ be a locally profinite group and $H$ its closed subgroup. It is well-known that a smooth representation of $G$ is identified with a smooth module over $\mathcal{H}(G)$, where $\mathcal{H}(G)$ ...

**4**

votes

**2**answers

234 views

### Ring of invariants for $n$-tuples of Lie algebras

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\M{M}\DeclareMathOperator\Tr{Tr}$Consider the diagonal action of $\GL(n,\mathbb{C})$ on the variety of $k$-tuples of matrices, $\M_{n\times n}(\mathbb{C}...

**3**

votes

**2**answers

72 views

### When representations of reductive Lie group in a Banach space and in its Garding space have the same length?

Let $G$ be a real reductive Lie group (e.g. $G=\operatorname{GL}(n,\mathbb{R})$). Let $\rho$ be a continuous representation of $G$ in a Banach space $V$. Let $V^\infty\subset V$ be the subspace of ...

**1**

vote

**1**answer

154 views

### The norm of the principal series intertwining operator for $\operatorname{GL}_2$

Is there a known bound on the norm of the standard intertwining operator for the principal series of $G = \operatorname{GL}_2(\mathbb Q_p)$?
Background:
For a character $\chi = (\chi_1,\chi_2)$ of the ...

**4**

votes

**1**answer

131 views

### Maximal torus of linear algebraic group over a ring

Let $G$ be a linear algebraic group over a $k$-algebra $A$, where $k$ is an algebraically closed field.
Consider the structure morphism $G\rightarrow U={\rm Spec}(A)$.
Assume that for every $k$-point ...

**1**

vote

**0**answers

46 views

### Maximal compact subgroups of p-adic orthogonal groups

Let F be a non-archimedean local field, and G be split orthogonal groups of odd degrees $\geq$ 3.
In this setting, my question is;
Is there explicit descriptions of maximal compact subgroups of G?
I ...

**3**

votes

**1**answer

62 views

### Calculating the residue of Eisenstein series from the residue of the intertwining operator

I've been reading the article Forms of $\operatorname{GL}(2)$ from the analytic point of view by Gelbart and Jacquet in Corvallis and am confused on a particular claim (equation 5.17 on page 232).
The ...

**4**

votes

**0**answers

47 views

### How to see the surjectivity of $L^2_{\text{cont}}$ onto the direct integral of Hilbert space representations?

I've been reading the article Forms of $\operatorname{GL}(2)$ from the analytic point of view by Gelbart and Jacquet in Corvallis and am confused on one point. Let $G = \operatorname{GL}_2$, and $V = ...

**9**

votes

**1**answer

165 views

### Why are characters orthogonal to cusp forms?

Let $G = \operatorname{GL}_2$, and let $V = L^2(Z(\mathbb A)G(\mathbb Q) \backslash G(\mathbb A),\omega)$, for $\omega$ a character of the ideles $\mathbb A^{\ast}$, identified with a central ...

**3**

votes

**0**answers

80 views

### Span of parabolic inductions of discrete series representations

Let $G$ be the $\mathbf{Q}_p$-points in a $p$-adic reductive group, and let $R(G)$ be the Grothendieck group of the category $\mathrm{Rep}(G)$ of finite-length admissible smooth complex ...

**3**

votes

**0**answers

73 views

### Localizations of smooth spherical varieties at simple roots

Setup
Let $G$ be a (connected) reductive group over an algebraically closed field $k$, and fix a Borel subgroup
$B \subset G$ and a maximal torus $T \subset B$. Let $\lambda: \mathbb{G}_m \to T$ be a ...

**5**

votes

**0**answers

77 views

### Problem with affine root subgroups of $SU_3$ ramified, residue characteristic $p=2$

Let $L/K$ be ramified quadratic extension of local fields, and let characteristic of the residue field of $K$ be $2$. Let $\mathbb{G}=SU_3$, $G=\mathbb{G}(K)$. Let $\text{val}$ be a valuation on $K$ ...

**3**

votes

**1**answer

69 views

### Image of square map on reductive group

$\DeclareMathOperator\ss{ss}$Let $G$ be a reductive group over a field $F$ of characteristic 0. (Here not necessarily $F=\overline{F}$.) Consider the square map
$$
G(F)\longrightarrow G(F), \quad g\...

**1**

vote

**0**answers

33 views

### Distributivity property for smooth parabolic induction

Let $G$ be a reductive group over a local field $k$ of characteristic zero with maximal split torus $T$, Weyl group $W$, Borel $B$ and a parabolic subgroup $P$ such that $P\supset B \supset T$. Denote ...

**2**

votes

**0**answers

69 views

### The kernel $K(x,y)$ as an integral over Eisenstein series for $\operatorname{GL}_2$

Let $G = \operatorname{GL}_2$, $f \in C_c^{\infty}(G(\mathbb A)/Z(\mathbb A))$, and $V = L^2( G(\mathbb Q)Z(\mathbb A)\backslash G(\mathbb A))$ (trivial central character). Then the operator $R(f)$ ...

**1**

vote

**0**answers

77 views

### $H(\mathbb A)^0/H(k)$ is homeomorphic to a closed set in $G(\mathbb A)/G(k)$

I'm reading Godement's paper Domaines fondamentaux des groupes arithmetiques and am confused about a part in a proof. The setting is $k$ is a number field, $\mathbb A$ is the ring of adeles of $k$, $...

**3**

votes

**1**answer

172 views

### Schubert cells in G/P for reductive G

All literature on the Schubert cells of the generalized flag varieties $G/P$ ("generalized" here means that $P$ is an arbitrary parabolic) assumes that $G$ is a semisimple complex group. I ...

**3**

votes

**2**answers

200 views

### Finiteness of the volume of $G(F) \backslash G(\mathbb A)$

Let $G$ be a semisimple algebraic group over a number field $F$ with trivial center. Let $\mathfrak S \subset G(\mathbb A)$ be a Siegel domain (defined in terms of a given maximal split torus and ...

**3**

votes

**0**answers

125 views

### Harmonic analysis on reductive groups over $\mathbb{R}$

A common way of doing harmonic analysis on (the $\mathbb{R}$-points of) reductive groups over $\mathbb{R}$ seems to be to use results from semisimple groups and "see what happens on the center&...

**3**

votes

**0**answers

117 views

### Centralizers of semisimple subgroups

$\DeclareMathOperator\GL{GL}$If $G$ is a simple Lie group, and $\rho: G \to \GL(V)$ is a representation, then by Schur's lemma, the group of automorphisms of $\rho$ is a reductive subgroup of $\GL(V)$....

**8**

votes

**1**answer

346 views

### Branching laws for smooth representations

Let $E / F$ be a quadratic extension of nonarchimedean local fields (characteristic 0 if it matters), and $\pi$ an irreducible infinite-dimensional smooth representation of $GL_2(E)$. Let $B$ be the ...

**2**

votes

**0**answers

47 views

### Conjugacy classes in centralizers

Let $G$ be a complex reductive group, let $g$ be an element, and let $C$ be the connected component of its centralizer. I'm curious about what is known about the intersection of conjugacy classes in $...

**4**

votes

**0**answers

60 views

### Is the union of conic orbits for a reductive group Zariski closed?

Let $G$ be a reductive group over an algebraically closed field $k$ of characteristic $p>0$. If $V$ is a rational $G$-module then we can define the Hilbert nullcone $\mathcal{N}(V)$ to be the zero ...

**1**

vote

**0**answers

48 views

### Relation between reductive homogeneous spaces and reductive groups

To start, I would like to note that my background on Lie algebras is quite basic, so this question might be trivial when seen from a Lie algebra perspective, which I lack.
We have the concept of a ...

**12**

votes

**2**answers

766 views

### Why are coroots needed for the classification of reductive groups?

As we know reductive groups up to isomorphism corresponds to root data up to isomorphism. My question is why in the definition of root data do we need the coroots?
Let's break it down to two questions:...

**2**

votes

**0**answers

83 views

### Representation of Levi subgroup $L\subset P \subset G$

Let $G$ be a split connected reductive group over a finite field extension of $\mathbb{Q}_p$ with split maximal torus $T$ of rank $d$ and simple roots $\Delta$. Furthermore associated to $I\subset \...

**6**

votes

**1**answer

202 views

### Real forms of complex reductive groups

I have a collection of related (to me) questions, which stem from the fact that I feel like I have a bunch of pieces, but not a full clear picture. I'm curious about forms of reductive groups in ...

**3**

votes

**1**answer

130 views

### Globalising tori and weak approximation

Let $G$ be a semi-simple and simply connected reductive group over $\mathbb{Q}$ and let $T \subset G_{\mathbb{Q}_p}$ be a maximal torus. A classical result of Harder tells us that we can find a ...

**2**

votes

**0**answers

80 views

### Determining a toric GIT quotient

Consider the following $G = (\mathbb{C}^*)^{\times 3}$-action on $\mathbb{A}^6$:
$(\lambda_0,\lambda_1,\lambda_2) \cdot (x_0,x_1,x_2,y_0,y_1,y_2) = (\lambda_0x_0,\lambda_1x_1,\lambda_2x_2,\frac{\...

**0**

votes

**1**answer

80 views

### The stabiliser group of an isotropic quadratic form over $\mathbb{Q}_p$ is non-compact?

Let $\mathbb{Q}_p$ denote the $p$-adic integers. Let $V$ be a $\mathbb{Q}_p$-vector space and $Q : V \rightarrow \mathbb{Q}_p$ be a non-degenerate integral quadratic form. We say that the pair $(Q,V)$ ...

**8**

votes

**1**answer

340 views

### Characterization of automorphic discrete spectra

I recently learned about automorphic spectral decomposition from the book "Spectral decomposition and Eisenstein series" by Moeglin and Waldspurger. (Let me call it M-W)
I have a question ...

**0**

votes

**0**answers

63 views

### Splitting of simply connected algebraic group

Let $k$ be a number field and let $G$ be a connected semisimple, simply connected algebraic group defined over $k$. Let $k'$ be a finite Galois extension over which $G$ splits. By the Chebotarev ...

**9**

votes

**1**answer

272 views

### Showing subgroups with equal Lie algebras are equal

Let $k$ be a field. It might as well be algebraically closed, but I do not want to assume that it has characteristic $0$. I will write "group" for "affine group scheme over $k$", ...

**7**

votes

**0**answers

95 views

### Chevalley restriction theorem: group vs lie algebra version

Let $G$ be a (split) reductive group over $k$, $T$ a split maximal torus, and W its Weyl group. I sometimes see the Chevalley restriction theorem stated as
(1) $k[G]^G \xrightarrow{\sim} k[T]^W$
and ...

**6**

votes

**1**answer

111 views

### Gelfand pair, weakly symmetric pair, and spherical pair

I am a bit confused with the relations among Gelfand pairs, weakly symmetric pairs, and spherical pairs defined in the book "Harmonic analysis on commutative spaces" written by professor ...

**3**

votes

**0**answers

133 views

### Tits Reductive Groups over Local Fields Example 1.15 (Quasi-split special unitary groups in odd dimension)

I hope this question about Tits's paper "Reductive groups over local fields" in Algebraic groups and discontinuous subgroups ends up having an easy answer, but I'm a little stuck on the ...

**4**

votes

**0**answers

74 views

### Slodowy slice intersecting a given orbit “minimally”?

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Is it true that for any $X\in\mathfrak{g}$, there exists an $\mathfrak{sl}_2$-triple $(e,h,f)$ in $\mathfrak{g}$ such that
We have $X\in e+Z_{\...

**2**

votes

**0**answers

54 views

### Reference request: boundedness for semistable principal bundles on a family of curves

We work over an algebraically closed field $k$.
Let $G$ be a reductive group and $X$ be a smooth projective curve over $k$. It is proven in [1, Theorem 1.2] that the moduli of semi-stable principal $G$...

**1**

vote

**0**answers

59 views

### Why does norm map the $\sigma$-conjugacy classes to the conjugacy classes?

Let $E/F$ be a cyclic extension of order $\ell$ (not assumed prime) of fields of characteristic $0$, and $\Sigma$ its Galois group; we denote by $\sigma$ a generator of $\Sigma$. We denote by $G(E), G(...

**2**

votes

**0**answers

75 views

### Bruhat-Tits theory: how does the normalizer act on an apartment?

Let $G$ be the points of a split, simply connected, semisimple algebraic group over a $p$-adic field $k$. Let $T$ be a maximal torus of $G$, $T_c$ its unique maximal compact subgroup, and $N$ the ...

**2**

votes

**0**answers

101 views

### Jordan normal form in a reductive group

Let $G$ be a connected complex reductive group. Given an element $g \in G$, there is a canonical decomposition $g = s u$ such that $s$ is semisimple, $u$ is unipotent and $s$ and $u$ commute. ...

**3**

votes

**0**answers

189 views

### Definition of Iwahori subgroup independently of the Bruhat-Tits building

Let $G$ be the points of a connected, semisimple algebraic group over a $p$-adic field $k$. To make life easy, let's assume the underlying group scheme is simply connected. The Bruhat-Tits building $...

**4**

votes

**0**answers

98 views

### Books on integration on semisimple Lie groups

Can anyone suggest me some good books where I can find integration theory on semisimple Lie groups (using KAK, KAN and other type of decompositions)?
I have read Knapp's book "Lie groups beyond ...

**3**

votes

**0**answers

106 views

### Reductive Lie groups and existence of maximal compact subgroup

I am reading Knapp's book "Lie groups beyond an introduction" (2nd edition). I am struggling to understand the following point. Recall that $G$ is a reductive Lie group. If the Lie algebra $\...