**5**

votes

**1**answer

153 views

### explicit integrals over a Lie group

I am looking for families of invariant integrals $\int_G dg f(g)$ (where $dg$ is a Haar measure) over a semisimple Lie group that can be evaluated in closed form, together with references where I can ...

**5**

votes

**1**answer

202 views

### Kernel of flux homomorphism (Calabi invariant) for volume-preserving maps on a compact manifold

Good morning everybody, I am currently reading through the book of Banyaga "Structure of classical diffeomorphism groups" link, and I am particularly interested in the question of factorizing ...

**16**

votes

**2**answers

719 views

### How bad can $\pi_1$ of a linear group orbit be?

Let $G$ be a simply connected Lie group and $\mathcal O= G(v)=G/G_v$ a $G$-orbit in some finite-dimensional $G$-module $V$. By the homotopy exact sequence, its fundamental group $\Gamma$ is the ...

**6**

votes

**1**answer

423 views

### Multiplicity one theorem

I am reading Dorian Goldfeld's book Automorphic forms and L functions for the groups GL(n,R) ...

**1**

vote

**1**answer

106 views

### Finite groups normalizing a torus

Let $G$ be a semi-simple linear algebraic group over the complex numbers, e.g. the special linear group. Can you find an example of a finite sub-group $H$ of $G$ which does not normalize any maximal ...

**0**

votes

**1**answer

90 views

### Dimension of Span of Adjoint orbit in $\mathfrak{su}(n)$

Given two elements $A,B \in \mathfrak{su}(n)$ what is the dimension of the span of the following adjoint orbit: $\{Ad_{e^{sA}}(B) \ | \ s \in [0,t]\}$ for different values of $t$. Does it ever change ...

**2**

votes

**0**answers

111 views

### What are the general zonal spherical functions for ${\rm SO}(n)/{\rm SO}(n-1)$?

The zonal spherical functions [1] on the sphere $(G={\rm SO}(n)$, $K={\rm SO}(n-1))$ are the Gegenbauer or ultraspherical polynomials if one considers the irreducible representations of ${\rm SO}(n)$ ...

**8**

votes

**5**answers

547 views

### Representation Theory of Lie Groups: Reference Request

I am looking for a reference that describes the correspondence between the (finite-dimensional) representations of (real) Lie groups and the representations of their Lie algebras. More precisely, ...

**1**

vote

**0**answers

60 views

### Largest dimensional Lie subgroup of $SU(N)$ [duplicate]

What is the largest (Lie) subgroup of $SU(n)$ in the sense of its dimension.
I am aware of this potential duplicate subgroup of SU(N) with maximal manifold dimension , however, the title of this ...

**5**

votes

**0**answers

120 views

### Is there a Lie II theorem for monoids?

Lie's second theorem says that if $G$ is a connected simply connected Lie group with Lie algebra $\mathfrak g$, then the functor of "differentiation" from the category $\mathrm{Rep}^f(G)$ of ...

**1**

vote

**0**answers

48 views

### Symplectic structures on the grassmannian model of the based loop group

$\newcommand{\Ad}{\operatorname{Ad}}$
In the study of (smooth/algebraic) based loop spaces of compact groups, one often uses a Grassmannian model to study the space. In particular, the Grassmannian ...

**2**

votes

**0**answers

134 views

### A homomorphism in the long exact sequence of a fibration for a homogeneous space of a Lie group

Let $G$ be a connected Lie group, and let $H\subset G$ be a (closed) Lie subgroup, not necessarily connected. Set $X=G/H$.
The fibration $j\colon G\to X$ with fiber $H$ induces an exact sequence
$$
...

**3**

votes

**1**answer

152 views

### Frame-bundle reduction from spinor-bundle reduction

Let $(M,g)$ be a $d$-dimensional Riemannian oriented, spin manifold, and let us denote by $F(M)$ its frame bundle, by $SP(M)$ its spin bundle and by $S = P(M)\times_{\rho}\Delta$ its spinor bundle, ...

**0**

votes

**0**answers

56 views

### Centralizer of a non-regular Lie algebra element

It is well understood that the centralizer of a regular element $A$ of a Lie algebra of complex (square, diagonalizable) matrices consists of polynomials $p(A)$ in that element of degree less than $n$ ...

**0**

votes

**0**answers

35 views

### rank of a Lie group over a non-archimedean local field of positive characteristic

In the case of a Lie algebra over a non-archimedean local field of positive characteristic (I have been led to believe that) it is not necessarily true that all Cartan subalgebras have the same ...

**2**

votes

**2**answers

348 views

### Local coordinates on (infinite dimensional) Lie groups, factorization of Riemann zeta functions

Given a (finite dimensional) Lie group $G$ (real $k=\mathbb{R}$ or complex $k=\mathbb{C}$) and its Lie algebra $\mathfrak{g}$, one can prove (a basis $B=(b_i)_{1\leq i\leq n}$ of $\mathfrak{g}$ being ...

**2**

votes

**1**answer

115 views

### Largest subgroup of $SU(n)$ for which the adjoint action preserves specific inner product on $\mathfrak{su}(N)$

Given $A,B \in \mathfrak{su}(n)$ such that $K(A, B)=0$, I am looking for the largest subgroup $H$ of $SU(n)$ for which:
$K \left(A, Ad_{U}(B) \right) = 0, \ \ \forall U \in H$ where $K$ is the ...

**2**

votes

**1**answer

117 views

### Number of connected components of the isometry group of a simply-connected lorentzian manifold

Let $(M,g)$ be a finite-dimensional connected lorentzian manifold. Then the group $G$ of isometries of $M$ (i.e., the group of diffeomorphisms $\varphi : M \to M$ with $\varphi^* g = g$) is a Lie ...

**5**

votes

**2**answers

223 views

### Is this characterization of (-1)-eigenspaces of the Weyl group of $E_6$ known?

I recently needed to know which circles $S$ in a maximal torus $T^6$ of the compact exceptional group $E_6$ yield one-dimensional subspaces $\mathfrak s$ of the Lie algebra $\mathfrak t^6$ that are ...

**4**

votes

**3**answers

271 views

### Reference request about the representations of the group $PSL_2(\mathbb{F}_q)$

Is there a review/exposition of the representation theory of $PSL_2(\mathbb{F}_q)$ ? Like an enumeration of its irreducible representations and their dimensions as a function of $q$?

**0**

votes

**1**answer

137 views

### Existence of a Lie algebra element orthogonal to the adjoint orbit of another element

Consider a compact, semi-simple, connected Lie group $G$ and its Lie algebra $\mathfrak{g}$. Denote the Killing form by $K$.
Given a single $A \in \mathfrak{g}$ when (i.e. which groups and which $A$) ...

**3**

votes

**1**answer

171 views

### Parallelizable nearly-Kahler manifolds

In this question, we have discussed how the following bundle:
$E_{d} = TS^{d}\oplus \Lambda^2 T^{\ast}S^{d}$
is always trivial, where $S^{d}$ is the $d$-dimensional standard sphere. Now, let us take ...

**3**

votes

**0**answers

205 views

### adding a boundary to the finite upper half-plane

Let $\Bbb{F}_q$ be a finite field, let $\delta \in \Bbb{F}_q$ be a non-square, let $\Bbb{F}_{q^2} = \Bbb{F}_q\big( \sqrt{\delta} \big)$ be the corresponding quadratic extension,
and let ...

**8**

votes

**1**answer

175 views

### finite upper half-plane model for the $\text{GL}_2(\Bbb{F}_q)$ Weil representation

Let $\Bbb{F}_q$ be a finite field with $q$ elements, let $\Bbb{F}_{q^2}$
be its quadratic extension, and consider the finite "upper" half space
${\frak{H}}_q := \Bbb{F}_{q^2} - \Bbb{F}_q$. Apeing a ...

**0**

votes

**0**answers

71 views

### determinants and principal series representations for $\text{GL}_2 \big( \Bbb{F}_q \big) $

Consider a finite field $\Bbb{F}_q$ and for simplicity let's assume $-1$ is not a square. Let $B$ be the Borel subgroup of $\text{GL}_2 \big( \Bbb{F}_q \big) $, for $i=1, 2$ let $\alpha_i:\Bbb{F}^* ...

**3**

votes

**0**answers

153 views

### regarding the upper half-plane model for the principal series representations of $\text{GL}_2\big( \Bbb{R}\big)$

Let $B$ be the Borel subgroup of $G = \text{GL}_2\big( \Bbb{R}\big)$, let
${\bf \alpha}:B \longrightarrow \Bbb{C}^*$ be a character, and consider the
induced representation $\text{Ind}_B^G ({\bf ...

**2**

votes

**0**answers

132 views

### Unitary representation of finite-dimensional unitary group

the question is the following. Let n,m be integers, $U(n)$ be the unitary group of $M_n(\mathbb C)$, and $\phi\colon U(n)\to U(m)$ be a continuous group homomorphism, that is moreover irreducible as a ...

**4**

votes

**1**answer

159 views

### Subgroups of $Sp_{2g}$ giving rise to Shimura data

Consider the Shimura datum $(GSp_{2g},\mathcal{H}_g)$. Let $G$ be a reductive $\mathbb{Q}$-subgroup of $Sp_{2g}$. I want to know under what condition there exists a point $x\in\mathcal{H}_g$ such that ...

**0**

votes

**0**answers

78 views

### Rationality of intersection of algebraic groups

Suppose that $G$ (defined over $\mathbb{Q}$) and $H$ (defined over $\mathbb{R}$) are two algebraic subgroups of a larger algebraic group defined over $\mathbb{Q}$. Assume that $G(\mathbb{R})$ and ...

**4**

votes

**1**answer

157 views

### Simply connected Lie groups homeomorphic to R^n are solvable

I have found the following claim in many proofs "Simply connected Lie groups homeomorphic to $\mathbb{R}^n$ are solvable". But the universal covering of $SL(2,\mathbb{R})$ satisfies the hypothesis of ...

**2**

votes

**1**answer

176 views

### Reading Ratner's paper “Ragunathan's conjectures for SL(2,R)”

Hello everyone (interested),
I am trying to read Marina Ratner's paper "Ragunathan's conjectures for $SL_{2}(R)$" (Israel Journal of Mathematics 80 (1992), 1-31). There is a claim right at the end of ...

**1**

vote

**0**answers

51 views

### $\Gamma$ cohomology of principal series

Let $G$ be a noncompact connected real semisimple Lie group with finited center. Let $\Gamma$ be a cocompact discrete subgroup of $G$, and let $P$ be a parabolique subgroup
with Langlands ...

**3**

votes

**1**answer

119 views

### Prescribed spherical representations, symplectic group $Sp(n)$

An irreducible representation $(\pi,V_\pi)$ of a compact group $G$ is called spherical with respect to the pair $(G,K)$, $K$ is closed subgroup of $G$, if $V_\pi$ has a non-zero vector invariant by ...

**17**

votes

**1**answer

276 views

### On a drawing in Dixmier's Enveloping Algebras

This image
comes from Dixmier's book, 'Enveloping Algebras' ('Algèbres enveloppantes').
Dixmier writes that
The curves shown on p. XIV have their origin in the study of U(sl(3)).
They are ...

**1**

vote

**0**answers

93 views

### What is the intersection of Spin(7) and U(4)?

I'm just curious from Berger's classification of Riemannian holonomy, how do Spin(7) manifolds intersect the other types of Riemannian manifolds?
In particular, what is the intersection of Spin(7) ...

**2**

votes

**0**answers

171 views

### Geometric proof of Borel-Weil theorem

I am curious if there is any geometric proof of Borel-Weil theorem.
Borel-Weil is a geometric realization of irreducible unitary representation. The proofs I found, however, all use Weyl unitarian ...

**0**

votes

**1**answer

202 views

### Suppose that $G$ is a subgroup of $GL_n(\mathbb C)$ with finite exponent. Then is $G$ a finite group? [closed]

As title. the exponent of $G$ is the least number $n$ (if exists) such that $g^n=e$ holds for all $g\in G$ or $+\infty$.

**1**

vote

**0**answers

146 views

### How the exceptional simple Lie groups/ algebras were first discovered and by whom?

I am wondering whether exceptional simple Lie groups/ algebras were first discovered in order to obtain a complete list of such objects, or they appeared as answers to completely different questions.
...

**1**

vote

**0**answers

56 views

### Computing equivariant K-theory using the amalgamted product

If I have a Lie group (or a Kac-Moody group) $G$ such that it's the amalgated product of it's proper parabolic subgroups $P_J$, i.e. $G = \text{colim} P_J$, then could I use this to compute the ...

**3**

votes

**0**answers

210 views

### Complex symplectic reduction

Oddly I find about zero resources talking about "complex symplectic reduction" upon a web search. Is there anything wrong with it?
I guess maybe there are two competing settings a priori: a complex ...

**0**

votes

**0**answers

68 views

### Bispinors, polyforms, bilinears and supersymmetric manifolds

Let $(V,q)$ be a regular quadratic vector space, and let us denote by $Cl(V,q)$ the corresponding Clifford algebra. Then there exists an isomorphism of $\mathbb{Z}_{2}$-graded algebras:
...

**9**

votes

**2**answers

210 views

### Neighborhoods of the identity in diffeomorphism groups

Finite dimensional Lie groups have the nice property that if $V$ is a small neighborhood of the identity, and $U \subset V$ another neighborhood, then $V$ is covered by $U^k$ (the set of all products ...

**2**

votes

**0**answers

130 views

### Kernel of the Weil homomorphism for compact symmetric spaces

Let $X = G/K$ be a Riemannian symmetric space of compact type and consider the "Weil homomorphism" $$w^\bullet: H^\bullet(BK; \mathbb R) \to H^\bullet(X; \mathbb R),$$ i.e. the map in cohomology ...

**0**

votes

**1**answer

68 views

### Lie group action on a finite dimensional flat manifold

Consider a finite dimensional flat Riemannian manifold $M$ quotiented by an action of a finite dimensional Lie group $G$, giving rise to the quotient $Q$.
First, assume that the action is isometric. ...

**4**

votes

**2**answers

212 views

### The action of the center on the extended Dynkin diagram

Let $R$ be an irreducible root system with a basis $\Pi$.
We obtain the Dynkin diagram $D$ and the extended Dynkin diagram ${\widetilde{D}}$ of $R$ with respect to $\Pi$.
Let $Q^\vee\subset P^\vee$ ...

**1**

vote

**0**answers

55 views

### Convention on Clifford Product

When studying the Clifford Algebra associated to some $(V,Q)$, one finds two basic identities differing by a sign:
$vv=Q(v)$ (see, for instance, Wikipedia)
$vv=-Q(v)$ (see, for instance, MathWorld ...

**2**

votes

**1**answer

85 views

### Bounds on Hilbert-Schmidt norm of difference of products of matrices

I suspect the following is well-known, but don't know of a reference (and it is not close to the area I normally work in).
I have two sequences of matrices $Q_{1},\ldots,Q_{k}$ and ...

**1**

vote

**1**answer

123 views

### $Spin(7)$ as stabilizer of a $4$-form revisited

For a better understanding of this question, please see the question and answer here.
In $Spin(8)$ there are plenty of copies of $Spin(7)$; consider, for instance, the antiimage of $SO(7)<SO(8)$ ...

**2**

votes

**0**answers

64 views

### Sections of inverse image sheaf of sheaf of sections of vector bundle

Let $\eta \colon Y \to Z$ be holomorphic map between complex manifolds, $E$ holomorphic vector bundle over $Z$, $\eta ^\ast E$ its pullback over $Y$, ${\cal O}_E$ sheaf of sections of $E$, and ...

**0**

votes

**1**answer

195 views

### generalization of highest weight theorem for semisimple lie algebras

Let $\mathfrak g$ be a real semisimple Lie algebra (without compact factors) with Iwasawa decomposition
$\mathfrak g=\mathfrak k\oplus \mathfrak a\oplus \mathfrak u$.
Let $\mathfrak p$ be a
...