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symplectic representations: when could the center act trivially?

I'm considering a problem about symplectic representation of real reductive group, which fits more or less into the setting of symplectic representations discussed in Milne's survey ''Shimura ...
turtle's user avatar
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2 answers
448 views

real orbits of highest weight vectors

Let $G_\mathbb{C}$ be a complex simple Lie group and let $V_\lambda$ be its finite dimensional irreducible representation with highest weight $\lambda$. Define $\mathcal{H}\_{\mathbb{C}} \subset V_\...
Vít Tuček's user avatar
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2 answers
1k views

Representation Theory of $U(N)$

(1) Is it true that the category of representations of $U(n)$ is equivalent to the category of representations of $SU(N) \times U(1)$? If so, how is it proved, or what is a good reference. (I guess ...
Andrea Pena's user avatar
0 votes
1 answer
210 views

Centers of universal enveloping algebra of complex Lie algebras

Let $\mathfrak{g}$ and $\mathfrak{g'}$ be complex Lie algebras such that $\mathfrak{g}$ is a subalgebra of $\mathfrak{g'}$. Let $Z(\mathfrak{g})$ and $Z(\mathfrak{g'})$ be the centers of the universal ...
Windi's user avatar
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1 answer
243 views

Adjoint action on the universal enveloping algebra and the PBW theorem

Let $\frak{g}$ be a semisimple Lie algebra and $U(\frak{g})$ its universal enveloping algebra. The adjoint action of $\frak{g}$ on itself extends to an action of $\frak{g}$ on $U(\frak{g})$. How does ...
Béla Fürdőház 's user avatar
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1 answer
205 views

What's an example of an $n$-step nilpotent Lie algebra whose centre is not $g^{(n)}$?

Let $\mathfrak g$ be a Lie algebra. $\mathfrak g^{(1)}=\mathfrak g$ and $\mathfrak g^{(n+1)}=[\mathfrak g,\mathfrak g^{(n)}]=\mathbb R$-span$\{[X,Y]:X\in\mathfrak g,Y\in\mathfrak g^{(n)}\}$. The ...
enihcamemit's user avatar
0 votes
1 answer
169 views

Representations of simply connected Lie groups [closed]

Let $G$ be a simply connected Lie group. Is it true that any finite dimensional representation of its Lie algebra is the differential of a representation of $G$? A reference would be helpful. Sorry if ...
asv's user avatar
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1 answer
130 views

Non-trivial weight spaces of finite-dimensional irreducible $\frak{g}$-modules

Let $\lambda \in \mathcal{P}^+$ be a dominant weight for $\frak{sl}(n,\mathbb{C})$. When does it hold that the zero weight space, of the associated finite-dimensional $L(\lambda)$, is non-trivial? ...
Martim Pereir's user avatar
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1 answer
194 views

Sum of weights of an irreducible representation of $U(N)$

Let $R$ be a finite-dimensional irreducible representation of $U(N)$, with the set of weights $W_R$. Each element of $W_R$ is a vector of length $N$ with integer entries. Firstly, I would like to know ...
Blind Miner's user avatar
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1 answer
274 views

when $g^*$ is invariant under $Ad(G)$?

Let $G$ be a Lie Group and $\mathfrak{g}$ be its lie algebra. Let $\mathfrak{g}$ is semisimple or reductive lie algebra, then prove that $\mathfrak{g}^*$ (dual of $\mathfrak{g}$)is invariant under $...
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1 answer
175 views

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 $\...
Windi's user avatar
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1 answer
187 views

Matrix representations of Lie groups of type $B_n$

For the Lie algebra $\mathfrak{so}(2n+1, \mathbb{C})$, there is a matrix representation given by the following matrices: \begin{align} \left( \begin{matrix} 0 & x & y \\ -y^T & A & B \\...
Jianrong Li's user avatar
  • 6,201
0 votes
1 answer
269 views

Quadratic Casimir of symplectic group

Does anyone know the formula for the value of quadratic Casimir of the symplectic group $Sp(2N)$ in the fundamental representation? In this definition, $Sp(2)=SU(2)$. Thanks a lot.
Mr. Gentleman's user avatar
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1 answer
791 views

Maximal subgroups of indefinite special orthogonal group SO(p,q)

Can someone answer the following question: Is there any classification of maximal proper Zariski-closed real subgroups of $SO(p,q)$ which are not parabolic, and satisfy the following conditions: ...
Vanya's user avatar
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1 answer
3k views

Why the Gell-Mann matrices in the SU(3)-model need to be trace orthogonal?

Why the Gell-Mann matrices in the SU(3)-model need to be trace orthogonal?
HAJV's user avatar
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0 answers
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A reference for this statement (representations of universal central extensions)

Let $G$ be a connected Lie group with universal cover $\tilde{G}$. I would really appreciate if someone could give me a reference for the proof of the following fact: "Every projective unitary ...
Mahtab's user avatar
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0 answers
74 views

A question on projective unitary representation of a Lie group

$\DeclareMathOperator\GL{GL}$Let $\mathcal{H}$ be a Hilbert space and $\GL(\mathcal{H})$ denote the group of invertible linear transformations of $\mathcal{H}$. Assume that $G=\{ f:\mathbb{P}\mathcal{...
Mahtab's user avatar
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0 answers
255 views

Any "inherent" definition of $\mathrm{SU}(2)$ independent of any matrix representation?

$\DeclareMathOperator\SU{SU}\SU(2)$ is explained in detail here. However, if I know right, this definition itself is known the "fundamental representation". I wonder if there is any "...
Isaac's user avatar
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0 answers
128 views

How to build a representation of the diffeomorphism group of $U(n)$?

Given that $U(n)$ is a smooth manifold I would like to know if there is a way of building a representation of $\text{Diff}(U(n))$ once you pick a particular (finite dimensional) representation of $U(n)...
Nicolas Medina Sanchez's user avatar
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0 answers
96 views

Integral of elements of random unitaries

It is known how to calculate the integral of elements of $N\times N$ Haar random unitaries using the Weingarten function: $$\int \prod_{k=1}^n U_{i_kj_k} U_{m_kr_k}^* \mathrm d U = \sum_{\sigma,\tau} \...
user50394's user avatar
  • 123
0 votes
0 answers
132 views

Lie algebra action Whittaker model

Let $(\pi, H)$ be an irreducible unitary generic representation of $G=\operatorname{GL}(r,\mathbb{C})$ and let $H^{\infty}$ be its subspace of smooth vectors. Let $W :G\to\mathbb{C} $ be the Whittaker ...
Akash Yadav's user avatar
0 votes
0 answers
228 views

How do I detect whether a representation is (or is not) the adjoint representation?

Let $(\mathfrak{g},[\cdot,\cdot])$ be a Lie algebra. There is a God-given representation of $\mathfrak{g}$, namely, the adjoint representation $\operatorname{ad} : \mathfrak{g} \to \operatorname{Der}(\...
AmorFati's user avatar
  • 1,379
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0 answers
79 views

$\tau$-admissible lift

I've been asked to take on a peer-review task which has to be completed in a short time, obviously details have to remain confidential, I need to work out what ``$\tau$-admissible lift" means if $...
Rupert's user avatar
  • 2,125
0 votes
0 answers
66 views

How to Evaluate the ABJM partition function for N=2

This is the ABJM partition function on the 3-sphere, $$ Z(2) = \int \frac{d^2\mu}{(2\pi)^2} \frac{d^2\nu}{(2\pi)^2} \frac{\left[ 2 \sinh \frac{\mu_1 - \mu_2}{2}\right]^2\left[ 2 \sinh \frac{\nu_1 - \...
john mangual's user avatar
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0 votes
0 answers
194 views

What are the E7(7) invariants in the adjoint representation?

Take a real vector space $R$ transforming in the adjoint representation of the ${\rm E}_7(7)$ Lie group as $R \rightarrow G R G^{-1}$. One can define invariants using traces of products of $R$ as ${\...
Geoffrey Compere's user avatar
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0 answers
53 views

Largest Set of Special Unitary Matricies With Invariant Subspace For Adjoint Action

I am trying to solve the following. Given the special unitary group $SU(n)$ and its adjoint action $Ad_{U}: \mathfrak{su}(n) \rightarrow \mathfrak{su}(n)$, what is the largest subset of $SU(n)$ such ...
Benjamin's user avatar
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0 answers
444 views

Cartan involutions of su(n)

I have a question regarding Cartan involutions of su(n). Some sources say that there is only one up to equivalence (Wikipedia on Cartan Decomposition). Others say there are Types I, II, III. I looked ...
magya_bloom's user avatar
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0 answers
272 views

minuscule representations and classical groups

Let $G$ a semisimple group over an algebraically closed field $k$. We assume that $G$ is classical. We call a $z$-extension, a group $\tilde{G}$ such that $\tilde{G}$ is a central extension of $G$ by ...
prochet's user avatar
  • 3,472
0 votes
0 answers
155 views

complex reductive Lie groups which are not defined over the real numbers

Hello Someone knows something about the complex reductive Lie groups (the complexification of a compact Lie group) which are not defined over the real numbers. I would like to know, if is it possible,...
R.Díaz's user avatar
-1 votes
1 answer
555 views

Representation of Lie algebra $\operatorname{SE}(2)$

When I read the paper Universal approximations of invariant maps by neural networks of Dmitry Yarotsky, it happens on page 36 that he used some concepts about the representation of Lie algebra of the ...
Chivul's user avatar
  • 129
-2 votes
1 answer
516 views

no classification of nilpotent lie groups

there is no classification of (simply connected) nilpotent lie groups, but I am tempted to try to generalize the construction of the Heisenberg group. For an upper triangular matrix: $$ \left( \...
john mangual's user avatar
  • 22.8k
-3 votes
1 answer
374 views

On Haar measure and Spherical measure [closed]

Let $d$-dimensional complex sphere be $$\{(c_1,\cdots,c_{d})\sum_{i=1}^{d} |c_i|^2=1.\}$$ We can define the Haar measure on this sphere by regarding the unitary group $U(d)$. We can regard the $d$-...
gondolf's user avatar
  • 1,503

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