5 problems

Let $(G_1,G_2)$ be a real reductive dual pair, let $\omega$ be the oscillator representation, and let $MG_1$ denote the metaplectic cover of $G_1$. For a representation $\pi$, its…

Let $\mathfrak{so}_{2n}$ and $\mathrm{O}_m$ act through the $(\mathfrak{so}_{2n},\mathrm{O}_m)$-duality on $\mathfrak{P}_{nm}$. On the $\mathrm{O}_m$ side, consider the cyclotomic…

Let $\mathfrak{l}$ be a Levi subalgebra of $\mathfrak{sp}_{2m}(\mathbb{C})$, and let $\nu^{(1)}$ and $\nu^{(2)}$ be dominant weights for $\mathfrak{l}$. Denote by the induced modul…

Let $L\subseteq L_1$ be a sublattice, let $A$, $B(L)$, $K_1$, $H_1(L)$, $L^\perp$, and $\psi_1^w$ be as defined in the lattice model, and let $r_2$ be the associated exponent. Seco…

Let $L\subseteq L_1$ be a sublattice, let $S_L$ be the associated subspace in the lattice model, and let $J_1(L)^\circ$ and $J_2(M)^\circ$ be the compact subgroups defined from the…