On a Conjecture of Barbasch and Pandžić

Abstract

Let G be a connected complex semisimple Lie group. Let J _s be the irreducible (𝔤, K) module with Zhelobenko parameters (ρ _c /2, − sρ _c /2), where s ∈ W is an involution. A conjecture of Barbasch and Pand\v zić claims that the Dirac cohomology of any unitary J _s is either zero or the trivial -type with multiplicity 2^{[l ₀/2]}, where l ₀ is the split rank of G. We prove this conjecture for J _s in the good range.

Key Words:

• Dirac cohomology
• Multiplicity one
• Spin lowest K-type

2010 Mathematics Subject Classification:

• 22E46

ACKNOWLEDGMENTS

Loke [5] told me that it follows from Theorem 2.1 of [8] that the representation J _{w 0} in Theorem 3.1 is K-multiplicity free. Lemma 3.3 was communicated to me by Lusztig [7]. I thank both of them sincerely. Finally, heartily gratitude is expressed to an anonymous referee for giving me many nice suggestions, which improve the quality of the paper a lot.

Notes

Communicated by A. Elduque.