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Unitary representations with Dirac cohomology: A finiteness result for complex Lie groups

Jian Ding and Chao-Ping Dong
From the journal Forum Mathematicum

Abstract

Let G be a connected complex simple Lie group, and let G^d be the set of all equivalence classes of irreducible unitary representations with non-vanishing Dirac cohomology. We show that G^d consists of two parts: finitely many scattered representations, and finitely many strings of representations. Moreover, the strings of G^d come from L^d via cohomological induction and they are all in the good range. Here L runs over the Levi factors of proper θ-stable parabolic subgroups of G. It follows that figuring out G^d requires a finite calculation in total. As an application, we report a complete description of F^4d.

MSC 2010: 22E46

Communicated by Freydoon Shahidi


Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11571097

Funding statement: The second-named author is supported by NSFC grant 11571097 and the China Scholarship Council.

A Appendix

In this appendix, we index all the involutions s in the Weyl group of F4 by presenting the weight sρ; see Table 17.

Table 17

Involutions in the Weyl group of F4.

IndexsρIndexsρIndexsρ
1[1,1,1,1]2[-1,2,1,1]3[2,-1,2,1]
4[1,3,-1,2]5[1,1,2,-1]6[-1,4,-1,2]
7[-1,2,2,-1]8[2,-1,3,-1]9[-1,-1,3,1]
10[5,-3,1,3]11[4,1,-2,4]12[1,5,-1,-1]
13[-1,-1,4,-1]14[-1,6,-1,-1]15[3,-5,3,3]
16[5,-1,-1,4]17[4,5,-4,2]18[-5,1,1,4]
19[-3,5,-3,5]20[7,-5,4,-2]21[6,1,1,-4]
22[-4,-1,2,4]23[-5,3,-1,5]24[-3,9,-5,3]
25[5,-7,6,-2]26[10,-5,1,1]27[7,-1,2,-4]
28[1,1,-3,7]29[6,3,-1,-3]30[-7,1,5,-3]
31[-5,7,1,-5]32[1,-4,1,6]33[9,1,-3,1]
34[-1,-3,1,6]35[-6,-1,6,-3]36[-2,1,-2,7]
37[-11,5,1,1]38[-7,5,3,-5]39[-5,9,-1,-4]
40[1,-2,-1,7]41[7,-10,4,3]42[10,-1,-2,1]
43[10,-3,1,-2]44[5,5,-7,5]45[9,1,-2,-1]
46[1,1,4,-8]47[-1,-1,-1,7]48[-9,11,-4,1]
49[1,-6,8,-6]50[10,-1,-1,-1]51[-7,-4,5,2]
52[-1,-5,8,-6]53[-11,9,-2,1]54[-11,7,1,-2]
55[-2,1,5,-8]56[-6,11,-7,4]57[-9,11,-3,-1]
58[5,-10,8,-4]59[1,-2,6,-8]60[9,-4,-3,6]
61[3,7,-3,-5]62[-1,-1,6,-8]63[-11,9,-1,-1]
64[1,-11,7,1]65[1,10,-8,1]66[-1,-10,7,1]
67[-5,-6,8,-3]68[-9,6,-4,6]69[-1,11,-8,1]
70[-4,11,-4,-4]71[4,-11,4,4]72[1,-11,8,-1]
73[9,-6,4,-6]74[5,6,-8,3]75[1,10,-7,-1]
76[-1,-10,8,-1]77[-1,11,-7,-1]78[11,-9,1,1]
79[1,1,-6,8]80[-3,-7,3,5]81[-9,4,3,-6]
82[-1,2,-6,8]83[-5,10,-8,4]84[9,-11,3,1]
85[6,-11,7,-4]86[2,-1,-5,8]87[11,-7,-1,2]
88[11,-9,2,-1]89[1,5,-8,6]90[7,4,-5,-2]
91[-10,1,1,1]92[-1,6,-8,6]93[9,-11,4,-1]
94[1,1,1,-7]95[-1,-1,-4,8]96[-9,-1,2,1]
97[-5,-5,7,-5]98[-10,3,-1,2]99[-10,1,2,-1]
100[-7,10,-4,-3]101[-1,2,1,-7]102[5,-9,1,4]
103[7,-5,-3,5]104[11,-5,-1,-1]105[2,-1,2,-7]
106[6,1,-6,3]107[1,3,-1,-6]108[-9,-1,3,-1]
109[-1,4,-1,-6]110[5,-7,-1,5]111[7,-1,-5,3]
112[-6,-3,1,3]113[-1,-1,3,-7]114[-7,1,-2,4]
115[-10,5,-1,-1]116[-5,7,-6,2]117[3,-9,5,-3]
118[5,-3,1,-5]119[4,1,-2,-4]120[-6,-1,-1,4]
121[-7,5,-4,2]122[3,-5,3,-5]123[5,-1,-1,-4]
124[-4,-5,4,-2]125[-5,1,1,-4]126[-3,5,-3,-3]
127[1,-6,1,1]128[1,1,-4,1]129[-1,-5,1,1]
130[-4,-1,2,-4]131[-5,3,-1,-3]132[1,1,-3,-1]
133[-2,1,-3,1]134[1,-2,-2,1]135[1,-4,1,-2]
136[-1,-1,-2,1]137[-1,-3,1,-2]138[-2,1,-2,-1]
139[1,-2,-1,-1]140[-1,-1,-1,-1]

Acknowledgements

The second-named author thanks the math department of MIT for offering excellent working conditions during October 2016 and September 2017. He is deeply grateful to the atlas mathematicians for many things, and to the referee for offering nice suggestions.

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Received: 2019-10-28
Revised: 2020-02-09
Published Online: 2020-03-19
Published in Print: 2020-07-01

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