Conjectural classification of scattered representations with Dirac cohomology

For a complex simple Lie group $G$, let $\widehat{G}^{\mathrm{sd}}$ denote the scattered part of the unitary dual with nonzero Dirac cohomology. Representations are encoded by the displayed tuples, and $[x]$ denotes the integer part of $x$; for admissible pairs $(a,b)$ define

$$p(a,b):=\max\{b-a-1,0\}.$$

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

• $\widehat{A}_{2n-1}^{\mathrm{sd}}$ is empty, while $\widehat{A}_{2n}^{\mathrm{sd}}$ consists of the $n$ representations

$$[\underbrace{2,\dots,2}_{p},\underbrace{1,\dots,1}_{2n-2p},\underbrace{2,\dots,2}_{p}],\qquad 0\leq p\leq n-1.$$

• $\widehat{B}_n^{\mathrm{sd}}$ consists of the $[n/2]$ representations

$$[\underbrace{2,\dots,2}_{p(a,b)},\underbrace{1,\dots,1}_{n-p(a,b)-1},2],$$

where $a+b=n$, $b\geq a\geq1$.

• $\widehat{C}_{2n-1}^{\mathrm{sd}}$ is empty, while $\widehat{C}_{2n}^{\mathrm{sd}}=\{\pi_{\mathrm{min}}\}$, where $\pi_{\mathrm{min}}$ is the minimal representation.
• $\widehat{D}_n^{\mathrm{sd}}$ consists of the $[n/4]$ representations

$$[\underbrace{2,\dots,2}_{p(a,b)},\underbrace{1,\dots,1}_{n-p(a,b)}],$$

where $a+b=n$, $b\geq a\geq2$, and $a$ is even.

In particular, any representation in $\widehat{G}^{\mathrm{sd}}$ is $K$-multiplicity free. This conjectural classification follows calculations in low-rank examples and gives an explicit description of the scattered representations for the classical types; its general status is not resolved in the supplied text.