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Suppose that $G$ is a connected reductive group over a non-archimedean local field $F$, and let $U\subset G$ be a unipotent subgroup with a nontrivial character $\psi: U\to \mathbb{C}^\times$. Also, l …

Let $\mathfrak{g}$ be a semi-simple Lie algebra over $\mathbb{C}$ with simply connected group $G$ and suppose that $$\mathfrak{g} = \bigoplus_i\mathfrak{g}_i$$ is a $\mathbb{Z}$- or $\mathbb{Z}/n\math …

$\DeclareMathOperator\GL{GL}$Let me try to clarify. The formula for the Whittaker functional in Theorem 4.6.5 of "Automorphic forms and representations" of Bump states that $$W\left(\pi\begin{pmatrix} …

$\DeclareMathOperator\Sym{Sym}$After further calculation, I realized that the answer to the first question is NO, the restriction of $\mathfrak g$ to $\mathfrak s$ need not be multiplicity free: suppo …

Yes, there are many relations between cluster algebras and crystal graphs. I am by no means an expert on these things, but let me mention one connection. Cluster algebras were originally discovered in …

A standard idea when dealing with the Arthur-Selberg trace formula (or a relative trace formula, for that matter) is to impose local conditions on the test function $f=\prod_vf_v$ to obtain a simple t …

Consider the Hall-Littlewood polynomial $$ P_\lambda(x_1,\ldots,x_n;t)=\sum_{\sigma\in S_n/S_n^\lambda}\sigma\left(x_1^{\lambda_1}\cdots x_n^{\lambda_n}\prod\limits_{\lambda_i>\lambda_j}\dfrac{x_i-tx_ …

I recently asked this question about Euler subgroups and generalizing the automorphic theory of $\mathrm{GL}_n$ to a more general setting. My question here is more specific. As mentioned there, the i …

For a general symmetric space (let’s say over $ \mathbb{C}$ for simplicity) to obtain a Cartan subspace, take a Cartan subalgebra $\mathfrak{t}$ of $\mathfrak{g}$ that is stable under the involution $ …

Recently, I have read about the Whittaker expansion for $\mathrm{GL}_n$ and was struck by the utility of the mirabolic subgroup, $\mathrm{P}_n\subset \mathrm{GL}_n$ of matrices with bottom row $(0\; 0 …

I am not entirely sure what you are asking for question 2, but let me take a crack at (1). At verious points I will be sloppy about distiguishing between a coset and its chosen representative, but it …

Consider an unramified quadratic extension $E/F$ of non-archimedean local fields, and suppose that $\langle\cdot,\cdot\rangle$ is a fixed Hermitian form on $E^d$ such that $\mathcal{O}_E^d$ is self-du …

Let $t=\varpi^\lambda$ where $\varpi$ is a uniformizer and $\lambda:\mathbb{G}_m\to T$ is a dominant weight. The assumption that $\lambda$ is dominant is harmless as we may conjugate by an appropriate …

Let ${G}$ be a simple algebraic group over $\mathbb{C}$ with maximal torus $T$ and set of simple roots $\{\alpha_i\}_{i\in \Delta}$. We then have a Borel supgroup $B=TU$ with unipotent radical $U$. Le …

Ngo famously proved the Langlands-Shelstad fundamental lemma for Lie algebras using the geometry of the Hitchin fibration. For the purposes of the trace formula, one actually needs the fundamental le …