32 problems

Let $G$ be a real reductive Lie group and let $X$ be a $(\mathfrak{g},K)$-module with Dirac cohomology $H_D(X)$. The infinitesimal character of $X$ is the character by which the ce…

Let $G$ be a complex simple group of type $E_8$, let $\widehat{G}^d$ denote its Dirac series, and let $K$ be a maximal compact subgroup of $G$. A representation…

Let $G$ be a real reductive group with Cartan decomposition $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$, and let $U(\mathfrak{g})$ and $C(\mathfrak{p})$ be the universal envelopi…

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 $M$ be a supermodule with an infinitesimal character, and let $H_D(M)$ denote its Dirac cohomology. Huang–Pandžić's superalgebraic Vogan conjecture. If $H_D(M)$ is nonzero, the…

Dong–Wong nonvanishing conjecture. If $A_\mathfrak{q}(\lambda)$ satisfies the H.P.-conditions but the displayed system has no solution in non-negative integers, then…

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…

Let $G$ be $Sp(2n,\mathbb{R})$ or $SO^*(2n)$, and consider the class of unitary representations constructed in Section 8 of Vogan's work, including all weakly fair…

Positivity conjecture. Proposition 2.1 holds for every $\pi\in\widehat{G}^d$.

Sharp multiplicity conjecture. One has

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…

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…

Huang's conjecture. For any irreducible $(\mathfrak{g},K)$-module $X$, there should be no cancellation between $H_D^+(X)$ and $H_D^-(X)$.

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

Huang's conjecture.

Let $M$ be an irreducible $(\mathfrak{g},K_\mathbb{R})$-module in a coherent family. Write its associated cycle as … where the multiplicities $m_i(M)$ extend to polynomials on…

Let $G(\mathbb{R})$ be a real reductive group, and let $\widehat{G(\mathbb{R})}^{\mathrm{d}}$ denote its irreducible unitary representations with nonzero Dirac cohomology. Let the…

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…

Let $G$ be a connected complex simple Lie group, and let $\widehat{G}^{\mathrm{d}}$ denote the set of irreducible unitary representations with non-zero Dirac cohomology. For repres…

For a weight module, the Euler–Poincaré pairing is defined by the alternating sum of the corresponding extension groups, while the Dirac index pairing is defined using Dirac indice…

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

Conjecture C. Any $\pi$ in the scattered part of $\widehat{G}^{\mathrm{d}}$ has a unique spin-lowest $K$-type, and this $K$-type must be u-small.

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 $W$ 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 $\pi$ 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…