**0**

votes

**0**answers

78 views

### Local coordinates on (infinite dimensional) Lie groups

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 ...

**0**

votes

**0**answers

28 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 $n$ ...

**5**

votes

**2**answers

181 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 ...

**0**

votes

**0**answers

21 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 ...

**11**

votes

**4**answers

796 views

### Which compact groups have finitely many irreducible representations of each dimension?

If my understanding is correct, this is true of sufficiently nice nonabelian Lie groups (see Ben Webster's answer below), and any finite group. On the other hand, this is false for any infinite ...

**2**

votes

**1**answer

93 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 ...

**0**

votes

**1**answer

126 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 $A$) ...

**2**

votes

**1**answer

86 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 ...

**3**

votes

**1**answer

150 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

**3**answers

225 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 $q$?

**3**

votes

**0**answers

191 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 ...

**9**

votes

**1**answer

1k views

### Is the Duflo polynomial conjecture open?

Let $G/K$ be a symmetric space. Let
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ be a Cartan decomposition,
with the odd part $\mathfrak{p}$. It is well known that the algebra of invariant
...

**0**

votes

**0**answers

16 views

### Is trace of regular representation in Lie group a delta function? [migrated]

My major is physics. I need to use some tools in group theory, but I am really
confused by the trace in compact infinite groups. The following is my question:
In discrete group theory, the ...

**8**

votes

**1**answer

157 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 ...

**16**

votes

**8**answers

895 views

### Examples of categorical adjunctions in analysis, Lie theory, and differential geometry?

In introductory texts on category theory, it seems like the majority of examples come from algebraic topology, algebra, and logic. Are there any good examples of adjunctions in analysis, Lie theory, ...

**0**

votes

**0**answers

66 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

146 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 ...

**8**

votes

**2**answers

673 views

### Group G hasn't all conditions of Lie group

Is there a group $G$ with the property that $G$ is a smooth manifold, the multiplication map of $G$ is smooth, but the inversion map of $G$ is not smooth?

**2**

votes

**0**answers

106 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

139 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

**1**answer

107 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
...

**0**

votes

**0**answers

76 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

140 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 ...

**3**

votes

**1**answer

98 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 ...

**2**

votes

**1**answer

167 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

48 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 ...

**17**

votes

**1**answer

257 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 ...

**3**

votes

**1**answer

358 views

### Is $M=E_{7(7)}/SU(7)\times\mathbb{R}^{+}$ a (pseudo)Kähler-Hodge manifold? Open problem

I have been told that the following is an open problem in mathematics, but I am pretty sure that experts in the topic surely know the answer.
Is the manifold
$$M=\frac{E_{7(7)}}{SU(7)}\times ...

**6**

votes

**6**answers

1k views

### Explicit equations for Schubert varieties

How can one compute the Schubert variety (by compute I mean having actual polynomials that define it) for SL(n)? If this is well known forgive my ignorance and just point me to the right book/paper.
...

**1**

vote

**0**answers

128 views

### representation of $SO(p)\times SO(q)$ with $p,q$ odd

Assume $p,q$ odd. We denote by $\sigma_p$ the standard representation of $SO(p)$, that is the representation of $SO(p)$ acting on $\mathbf{R}^p$ as matrix. So is $\sigma_q$.
Take $K=SO(p)\times ...

**1**

vote

**0**answers

86 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) ...

**3**

votes

**0**answers

184 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 ...

**2**

votes

**0**answers

152 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 ...

**10**

votes

**1**answer

259 views

### Derivation of Blattner's conjecture in the Beilinson-Bernstein picture

On the last page of Schmid's article "Discrete Series", he says
"In the Beilinson-Bernstein picture, discrete series modules are attached to closed $K$-orbits in $X$... the $K_{\mathbb R}$-structure ...

**9**

votes

**2**answers

193 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 ...

**0**

votes

**1**answer

195 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

140 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

52 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 ...

**10**

votes

**2**answers

586 views

### Groups “approximately commutative” near the identity

Is the following idea something that is known?
I call a metric group[1] $G$ "approximately commutative near the identity" if there exists a $K$ such that for small enough $\epsilon$, when $d(g,id) ...

**4**

votes

**2**answers

193 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 $Q^\vee\subset P^\vee$ ...

**0**

votes

**0**answers

62 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:
...

**2**

votes

**0**answers

123 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 ...

**1**

vote

**0**answers

112 views

### Finding density of Haar measure relative to the Liouville volume measure on $\frac{G^{\mathbb{C}}}{P}$

Let $G$ be a connected compact Lie group and $G^{\mathbb{C}}$ be the complexification of the Lie group $G$ then we know that by polar decomposition we can write $G^{\mathbb{C}}\cong G\times ...

**1**

vote

**0**answers

205 views

### Orbital integral by using symplectic quotient

Let $(M,\omega)$ be a compact symplectic Hamiltonian $G$-manifold and $\Gamma_{\text hol}(M,L)$ be the space of holomorphic sections of the line bundle $L\to M$
I am looking for a proof for ...

**0**

votes

**1**answer

61 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. ...

**1**

vote

**0**answers

48 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 ...

**4**

votes

**0**answers

263 views

### distributions on Lie groups and representations

Let $G$ be a Lie group and $\pi$ a continuous action on $V$, a Fréchet space.
This action induces a representation of the space of compactly supported functions, $C_c(G)$, with convolution as product ...

**2**

votes

**1**answer

71 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

111 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

53 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 ...