29 problems

Let $V$) be a simple vertex operator algebra satisfying the following conditions: $V_{(0)}=\mathbb{C}\mathbf{1}$, $V_{(n)}=0$ for $n<0$, and the contragredient $V'$ is equivalent t…

Let $\Phi^{\vec v}_{\vec u}(w):\mathcal{F}_{\vec u}\to\mathcal{F}_{\vec v}$ be the normalized intertwining operator, and let $|K_{\vec\mu}\rangle$ and $\langle K_{\vec\lambda}|$ be…

Symmetrized-kernel conjecture. The measure $\mu$ obtained using these symmetrized kernels is the correct measure in the integral representation of the intertwining operator.

Trace-realization conjecture. The function $u$ is obtained by taking a similar trace of products of intertwining operators for the quantum affine algebra.

Frenkel–Reshetikhin conjecture. Taking traces of products of intertwiners, rather than matrix elements, should produce solutions to the elliptic $r$-matrix equations.

Spectral-automorphic intertwiner conjecture. Under the relative Langlands equivalences, $I_{\mathrm{aut}}$ is equivalent to the spectral push-pull functor $I_{\mathrm{spec}}$ along

Normalized intertwining-cokernel conjecture. One has

Zelevinsky pole conjecture. One has

Intertwining-pole conjecture. One has

Let $(G,H)$ be a unimodular reductive symmetric pair over a local field of characteristic zero. Let $P=MU$ be a parabolic subgroup of $G$, let $\sigma$ be an irreducible representa…

Let $V\in\mathcal K_0$ and let $H=-\Delta+V$. For the intertwining operators involving powers of $H$ and $-\Delta$, consider their boundedness on $L^p$. Optimal-range conjecture. A…

Duality conjecture. There exists a maximally extended multivalued analytic function $f(z_1,z_2;w_1,w_2,w_3,w_4)$ on

Moore–Seiberg conjecture. For every $n\in\mathbb{N}$, the space spanned by shifted $q$-traces of products of $n$ chiral vertex operators is invariant under modular transformations.

Whittaker-space isomorphism conjecture. For every $W$-orbit $\mathcal{O}\subset\mathscr{X}_{Q,n}$, the embedding is an isomorphism

Let $M^*_{c,a}(s,\tau,\sigma_{\bar{r}})$ be the normalized intertwining operator introduced above, with $s\in\mathbb{C}$. Nonvanishing conjecture. The operator…

Casselman–Shahidi's conjecture. The normalized standard intertwining operator $M^*(s,\tau_a,\sigma_\gamma)$ is holomorphic in $s\in\mathbb{C}$. This conjecture proposes that the no…

Strong nonvanishing conjecture. The operator $M^*(s,\rho_c(\tau_a)\otimes \rho_d(\tau_b))$ is nonzero for every $s\in\mathbb{C}$.

Twisted intertwining-operator conjecture. The twisted intertwining operators among the $g$-twisted $V$-modules, for all $g\in G$, satisfy associativity, commutativity and modular i…

Intertwining-operator conjecture. For all $1<l\leq r$, $M(\zeta,w_{(k^{l})})$ is well defined at $\zeta=\zeta^{(l)}$ and has irreducible image. For all $c\geq1$,…

Compression conjecture. Any intertwining operator of $V_{\mathfrak g}^l$ can be realized as the compression of an intertwining operator of…

Let $R$ be a root system and let $k$ and $k^\tri$ be multiplicities on $R$, with $k^\tri>k>0$. Write $V_{k^\tri,k}=V_{k^\tri}\circ V_k^{-1}$ for the intertwining operator between t…

Let $F$ be the relevant number field, let $\nu$ be a place of $F$, and let $\tau_\nu$ be the local representation used to form the induced representations. For $l\geq 2$, write…

For a reductive group $G$ over a local field, let property (BDmax) mean that there is a constant $C>0$ such that, for every maximal parabolic subgroup $P=MU$, every smooth irreduci…

Let $\mathbb{S}$ denote the spinor representation, and consider differential intertwining operators … with integral kernels…

Let $\widetilde{H}$ be the subgroup acting on spinor-valued principal series representations, and let the previously constructed spinor-valued differential intertwining operators b…