On the space of 𝐾-finite solutions to intertwining differential operators

Kubo, Toshihisa; Ørsted, Bent

doi:10.1090/ert/527

On the space of $K$-finite solutions to intertwining differential operators
HTML articles powered by AMS MathViewer

by Toshihisa Kubo and Bent Ørsted

Represent. Theory 23 (2019), 213-248

DOI: https://doi.org/10.1090/ert/527

Published electronically: September 10, 2019

PDF | Request permission

Abstract:

In this paper we give Peter–Weyl-type decomposition theorems for the space of $K$-finite solutions to intertwining differential operators between parabolically induced representations. Our results generalize a result of Kable for conformally invariant systems. The main idea is based on the duality theorem between intertwining differential operators and homomorphisms between generalized Verma modules. As an application we uniformly realize on the solution spaces of intertwining differential operators all small representations of $\widetilde {\mathrm {SL}}(3,\mathbb {R})$ attached to the minimal nilpotent orbit.

References

George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
L. Barchini, Anthony C. Kable, and Roger Zierau, Conformally invariant systems of differential equations and prehomogeneous vector spaces of Heisenberg parabolic type, Publ. Res. Inst. Math. Sci. 44 (2008), no. 3, 749–835. MR 2451611, DOI 10.2977/prims/1216238304
L. Barchini, Anthony C. Kable, and Roger Zierau, Conformally invariant systems of differential operators, Adv. Math. 221 (2009), no. 3, 788–811. MR 2511038, DOI 10.1016/j.aim.2009.01.006
I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, Structure of representations that are generated by vectors of highest weight, Funkcional. Anal. i Priložen. 5 (1971), no. 1, 1–9 (Russian). MR 0291204, DOI 10.1007/BF01075841
I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, Differential operators on the base affine space and a study of $\mathfrak {g}$-modules, Lie Groups and their Representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), 1975, pp. 21–64.
B. Binegar and R. Zierau, Unitarization of a singular representation of $\textrm {SO}(p,q)$, Comm. Math. Phys. 138 (1991), no. 2, 245–258. MR 1108044, DOI 10.1007/BF02099491
B. Binegar and R. Zierau, A singular representation of $E_6$, Trans. Amer. Math. Soc. 341 (1994), no. 2, 771–785. MR 1139491, DOI 10.1090/S0002-9947-1994-1139491-1
David H. Collingwood and Brad Shelton, A duality theorem for extensions of induced highest weight modules, Pacific J. Math. 146 (1990), no. 2, 227–237. MR 1078380, DOI 10.2140/pjm.1990.146.227
Jacques Dixmier, Algèbres enveloppantes, Cahiers Scientifiques, Fasc. XXXVII, Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1974 (French). MR 0498737
Jose A. Franco and Mark R. Sepanski, Global representations of the heat and Schrödinger equation with singular potential, Electron. J. Differential Equations (2013), No. 154, 16. MR 3084634
N. Hashimoto, K. Taniguchi, and G. Yamanaka, The socle filtrations of principal series representations of ${SL}(3,\mathbb {R})$ and ${S}p(2,\mathbb {R})$, preprint, arXiv:1702.05836.
J. Hilgert, A. Pasquale, and T. Przebinda, Resonances for the Laplacian on Riemannian symmetric spaces: the case of $\mathrm {SL}(3,\Bbb {R})/\mathrm {SO}(3)$, Represent. Theory 21 (2017), 416–457. MR 3710652, DOI 10.1090/ert/506
James E. Humphreys, Representations of semisimple Lie algebras in the BGG category $\scr {O}$, Graduate Studies in Mathematics, vol. 94, American Mathematical Society, Providence, RI, 2008. MR 2428237, DOI 10.1090/gsm/094
Markus Hunziker, Mark R. Sepanski, and Ronald J. Stanke, The minimal representation of the conformal group and classical solutions to the wave equation, J. Lie Theory 22 (2012), no. 2, 301–360. MR 2976923
Markus Hunziker, Mark R. Sepanski, and Ronald J. Stanke, A system of Schrödinger equations and the oscillator representation, Electron. J. Differential Equations 260 (2015), 28 pp.
Anthony C. Kable, $K$-finite solutions to conformally invariant systems of differential equations, Tohoku Math. J. (2) 63 (2011), no. 4, 539–559. MR 2872955, DOI 10.2748/tmj/1325886280
Anthony C. Kable, Conformally invariant systems of differential equations on flag manifolds for $G_2$ and their $K$-finite solutions, J. Lie Theory 22 (2012), no. 1, 93–136. MR 2859028
Anthony C. Kable, The Heisenberg ultrahyperbolic equation: $K$-finite and polynomial solutions, Kyoto J. Math. 52 (2012), no. 4, 839–894. MR 2998915, DOI 10.1215/21562261-1728911
Anthony C. Kable, The Heisenberg ultrahyperbolic equation: the basic solutions as distributions, Pacific J. Math. 258 (2012), no. 1, 165–197. MR 2972482, DOI 10.2140/pjm.2012.258.165
Toshiyuki Kobayashi and Michael Pevzner, Differential symmetry breaking operators: I. General theory and F-method, Selecta Math. (N.S.) 22 (2016), no. 2, 801–845. MR 3477336, DOI 10.1007/s00029-015-0207-9
Toshiyuki Kobayashi and Bent Ørsted, Analysis on the minimal representation of $\mathrm O(p,q)$. I. Realization via conformal geometry, Adv. Math. 180 (2003), no. 2, 486–512. MR 2020550, DOI 10.1016/S0001-8708(03)00012-4
Toshiyuki Kobayashi and Bent Ørsted, Analysis on the minimal representation of $\mathrm O(p,q)$. II. Branching laws, Adv. Math. 180 (2003), no. 2, 513–550. MR 2020551, DOI 10.1016/S0001-8708(03)00013-6
Toshiyuki Kobayashi and Bent Ørsted, Analysis on the minimal representation of $\mathrm O(p,q)$. III. Ultrahyperbolic equations on ${\Bbb R}^{p-1,q-1}$, Adv. Math. 180 (2003), no. 2, 551–595. MR 2020552, DOI 10.1016/S0001-8708(03)00014-8
A. Korányi and H. M. Reimann, Equivariant first order differential operators on boundaries of symmetric spaces, Invent. Math. 139 (2000), no. 2, 371–390. MR 1738448, DOI 10.1007/s002229900030
Bertram Kostant, The vanishing of scalar curvature and the minimal representation of $\textrm {SO}(4,4)$, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989) Progr. Math., vol. 92, Birkhäuser Boston, Boston, MA, 1990, pp. 85–124. MR 1103588
Toshihisa Kubo, A system of third-order differential operators conformally invariant under $\mathfrak {sl}(3,\Bbb C)$ and $\mathfrak {so}(8,\Bbb C)$, Pacific J. Math. 253 (2011), no. 2, 439–453. MR 2878818, DOI 10.2140/pjm.2011.253.439
J. Lepowsky, Uniqueness of embeddings of certain induced modules, Proc. Amer. Math. Soc. 56 (1976), 55–58. MR 399195, DOI 10.1090/S0002-9939-1976-0399195-4
Adam R. Lucas, Small unitary representations of the double cover of $\textrm {SL}(m)$, Trans. Amer. Math. Soc. 360 (2008), no. 6, 3153–3192. MR 2379792, DOI 10.1090/S0002-9947-08-04401-2
Bent Ørsted, Generalized gradients and Poisson transforms, Global analysis and harmonic analysis (Marseille-Luminy, 1999) Sémin. Congr., vol. 4, Soc. Math. France, Paris, 2000, pp. 235–249 (English, with English and French summaries). MR 1822363
T. Oshima, An elementary approach to the Gauss hypergeometric function, Josai. Math. Monogr. 6 (2013), 3–23.
John Rawnsley and Shlomo Sternberg, On representations associated to the minimal nilpotent coadjoint orbit of $\textrm {SL}(3,\,\textbf {R})$, Amer. J. Math. 104 (1982), no. 6, 1153–1180. MR 681731, DOI 10.2307/2374055
Mark R. Sepanski and Jose A. Franco, Global representations of the conformal group and eigenspaces of the Yamabe operator on $S^1\times S^n$, Pacific J. Math. 275 (2015), no. 2, 463–480. MR 3347378, DOI 10.2140/pjm.2015.275.463
Dj. Šijački, The unitary irreducible representations of $\overline {\textrm {SL}}(3,\,R)$, J. Mathematical Phys. 16 (1975), 298–311. MR 383989, DOI 10.1063/1.522541
H. Tamori, Minimal representations of $\widetilde {SL}(3,\mathbb {R})$ and $\widetilde {O}(3,4)$, Master’s thesis, the University of Tokyo, 2017.
Pierre Torasso, Quantification géométrique, opérateurs d’entrelacement et représentations unitaires de $(\widetilde \textrm {SL})_3(\textbf {R})$, Acta Math. 150 (1983), no. 3-4, 153–242 (French). MR 709141, DOI 10.1007/BF02392971
Wan-Yu Tsai, Some genuine small representations of a nonlinear double cover, Trans. Amer. Math. Soc. 371 (2019), no. 8, 5309–5340. MR 3937294, DOI 10.1090/tran/7351
Daya-Nand Verma, Structure of certain induced representations of complex semisimple Lie algebras, Bull. Amer. Math. Soc. 74 (1968), 160–166. MR 218417, DOI 10.1090/S0002-9904-1968-11921-4
David A. Vogan Jr., Associated varieties and unipotent representations, Harmonic analysis on reductive groups (Brunswick, ME, 1989) Progr. Math., vol. 101, Birkhäuser Boston, Boston, MA, 1991, pp. 315–388. MR 1168491
Wei Wang, Representations of $\textrm {SU}(p,q)$ and CR geometry. I, J. Math. Kyoto Univ. 45 (2005), no. 4, 759–780. MR 2226629, DOI 10.1215/kjm/1250281656