15 problems

Let $G$ be a complex classical Lie group, and let a unitary representation of $G$ be given. A unitary representation has nonzero Dirac cohomology if its Dirac cohomology does not v…

Barbasch–Pandžić conjecture. If $\pi\in\widehat{G}$ has regular, half-integral infinitesimal character, then

Let $W$ be the reflection group, let $\mathfrak{h}$ be its reflection representation, let $\widetilde{W}$ be the associated pin double cover, and let $\chi$ be the spin module char…

Let ${\mathfrak g}$ be a Lie superalgebra, ${\mathfrak l}$ a subalgebra, $M$ a ${\mathfrak g}$-supermodule, and $x\in {\mathcal Y}_{{\mathfrak l}}$. Write…

Let $G$ be an equal-rank real reductive group with maximal compact subgroup $K$, and let $\widetilde K$ be the spin double cover of $K$. For an irreducible unitary…

Barbasch–Pandžić multiplicity-one conjecture. For every $\pi\in\widehat{G}^d$, there is a unique spin-lowest $K$-type $V_{\mathfrak{k}}(\eta)$, and it occurs in $\pi$ with multipli…

Dirac-series conjecture. Any Dirac series member of $U(p,q)$ must be a weakly fair $A_{\mathfrak{q}}(\lambda)$ module satisfying $$ .

Let $N_{A_n}$ denote the cardinality of the scattered part of $\widehat{A}_n^{\mathrm{d}}$, the set of irreducible unitary representations for the complex simple group of type…

Scattered-representation classification conjecture. The set $\widehat{G}^{\mathrm{sd}}$ can be described as follows:

Let $\overline{{\bf H}}$ be the restricted rational Cherednik algebra, let $\mathfrak{h}$ be the reflection representation of $W$, let $\widetilde{W}$ be the associated pin double…

Let $\pi$ be a unitary representation. A representation is induced by parabolic induction when it is obtained from a representation of a parabolic subgroup in this way. Unitary Dir…

Let $G(\mathbb{R})\supset K(\mathbb{R})$ be a real reductive group, let $\pi$ be an irreducible admissible representation, and let $X_\pi$ be its Harish-Chandra module. A represent…

Let $X$ be an irreducible $(\mathfrak{g},K)$-module, and let $H_D^+(X)$ and $H_D^-(X)$ be the even and odd parts of its Dirac cohomology. Dirac cohomology parity conjecture. … A re…

Let $\widetilde K$ be the spin cover of a maximal compact subgroup $K$, and let $\mathsf{I}_{\mathrm{Dir}}(X)$ denote the Dirac index of a finite-length Harish-Chandra module $X$.…

Multiplicity conjecture. A unitary representation with infinitesimal character $\rho/2$ has Dirac cohomology consisting of the trivial $K$-type with multiplicity…