systems and harmonic analysis) Citation 数理解析研究所講究録 (2002), 1294:

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1 Enbeddings of derived functor modul Titleprincipal series (Representations o systems and harmonic analysis) Author(s) Matumoto Hisayosi Citation 数理解析研究所講究録 (2002) 1294: Issue Date URL Right Type Departmental Bulletin Paper Textversion publisher Kyoto University

2 $\mathrm{m}$ is : : to Enbeddings of derived functor modules into degenerate principal series Hisayosi Matumoto ( ) Graduate School of Mamatical Sciences University of Tokyo Komaba Tokyo JAPAN ( ) hisayosi@msu-tokyoacjp \S 1 Formulation of problem Let be areal linear reductive Lie group $G$ $G\mathbb{C}$ $\mathrm{g}_{0}$ and let its complexi\s cation We denote by (resp 9) Lie algebra of (resp $G$ $Gq$ $\sigma$ ) and denote by complex conjugation on 9with respect to go We \S x amaximal compact subgroup of and denote by $I\{ $ $G$ $\ta$ corresponding $\mathrm{f}$ $I\acute{\mathrm{t}}$ Cartan involution We denote by complexi\s ed Lie algebra of We \S x aprabolic subgroup of $P$ with $\ta$-stable Levi $G$ part $M$ We denote by $N$ nilradical of We denote by $P$ $\mathfrak{p}$ $\mathrm{m}$ $\mathrm{n}$ and complexi\s ed Lie algebras of $P$ $M$ and $N$ $M\mathbb{C}$ $N\mathbb{C}$ $\mathfrak{p}$ respectively We denote by Pc and analytic subgroups in $Gc$ with respect to $\mathfrak{n}$ and respectively For we de\s ne $X\in \mathrm{m}$ $\delta(x)=\frac{1}{2}\mathrm{t}\mathrm{r}(\mathrm{a}\mathrm{d}_{g}(x) _{\mathfrak{n}})$ $\delta$ $2\delta$ $\mathrm{m}$ Then aone-dimesional representation of We see that lts to aholomorphic group $arrow \mathbb{c}^{\mathrm{x}}$ $\xi_{2\delta}$ $\xi_{2\delta} _{N_{\mathbb{C}}}$ homomorphism $NIc$ $\xi_{2\delta}$ De\S ning trivial we may extend to $Pc$ We put $X=Gc/Pc-$ Let be holomorphic line bundle on $L$ $X$ corresponding to canonical divisor Namely is $G\mathbb{C}$-homogeneous line bundle on $C$ $X$ $\xi_{2\delta}$ associated to character $P\mathbb{C}$ $\xi_{2\delta}$ on We denote restriction of $P$ by same letter $Parrow \mathbb{c}^{\mathrm{x}}$ $u1\mathrm{n}\mathrm{d}_{p}^{g}(\eta)$ $\eta$ For acharacter we consider unnormalized parabolic induction $u1\mathrm{n}\mathrm{d}_{p}^{g}(\eta)$ $C^{\infty}$ Namely is $\mathrm{k}$-\s nite part of space of -sections of G-homogeneous $u1\mathrm{n}\mathrm{d}_{p}^{g}(\eta)$ line bundle on $G/P$ $\eta$ associated to is aharish-chandra -module $(\mathrm{g} K)$ If $G/P$ is orientable n trivial $G$ -representation is unique irreducible quotient $u1\mathrm{n}\mathrm{d}_{p}^{g}(\xi_{2\delta})$ of If $G/P$ $\omega$ is not orientable re is acharacter on $P$ $\omega$ such that is trivial on identical componnent of $P$ and trivial $G$ -representation is unique irreducible $u\mathrm{i}\mathrm{n}\mathrm{d}_{p}^{g}(\xi_{2\delta}\otimes\omega)$ quotient of Let be an open -orbit on $O$ $G$ $X$ We put following assumption

3 (resp satisfies is 73 Assumption 11 There is a $\ta$-stable parabolic subalgebra q of g such that q $\in O$ Under above assumption q has alevi decompsition q such that $=[+\mathrm{u}$ $\mathfrak{l}$ $\ta$ a and part In fact [is $\mathfrak{l}=\sigma(\mathrm{q})\cap \mathrm{q}$ unique since we have For each open -orbit O on $G$ X we put $\mathrm{c}\mathrm{r}$-stable Levi $A_{\mathcal{O}}=\mathrm{H}^{\dim u\cap \mathrm{t}}(o \mathcal{l})_{k- \mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}}$ Namely in termilogy in [Vogan-Zuckerman 1984] we have $A_{\mathcal{O}}=A_{\mathrm{q}}=A_{\mathrm{q}}(0)$ We consider following problem: $A_{\mathcal{O}}\mapsto u1\mathrm{n}\mathrm{d}_{p}^{g}(\xi_{2\delta})$ Problem 12 Is re an embedding: or \S 2 Complex groups Let $G$ $Aoarrow\epsilon 1u\mathrm{n}\mathrm{d}_{P}^{G}(\xi_{2\delta}\otimes\omega)$? be aconnected real split reductive linear Lie group Here we consider Problem 12 for complexication $Gc$ rar than itself Embedding $G$ $G\mathbb{C}$ into $Gc$ $\cross Gc$ via $g[]$ $(g \sigma(g))$ $G_{\mathbb{C}}\cross G\mathbb{C}$ $G_{\mathbb{C}}$ we may regard as acomplexication of Each parabolic subgroup of $Gc$ is complexication of aparabolic subgroup of $G$ Let $P$ be aparabolic subgroup of $G$ Then complexication of $Pc$ $P\mathbb{C}\cross P\mathbb{C}$ can be identied with via above embedding $G\mathbb{C}\mapsto G_{\mathbb{C}}\cross Gc-$ Hence complex generalized flag variety for $Gc$ is $X\cross X$ $\ta$ We fix a and $\subseteq \mathfrak{p}$ $\sigma$-stable Cartan subalgebra [$)$ of 9such that [$)$ We denote by $w_{0}$ $w_{\mathfrak{p}}$ (resp ) longest $(\mathrm{g} $\mathfrak{h}$ $\mathrm{m}$ element of Weyl group with respect to ( )) We easily have: Proposition 21 $X\cross X$ has a unique $G_{\mathbb{C}}$ and only $w_{0}w_{\mathfrak{p}}=w_{\mathfrak{p}}w_{0}$ $\mathcal{o}_{\mathbb{c}}$ -orbit(say Oq) We consider $ \xi_{2\delta}$ for Then character $G$ $\xi_{2\delta}\mathrm{h}$ $\nu$ $P\mathbb{C}$ For characters $\mu$ $\mu and of we denote restriction of of $Pc$ $\cross Pc$ as above by same letter For complex case we have : Theorem 22 ([Vogan-Zuckerman1984]) $\xi_{2\delta}$ on Assumption 11 $Gc$ $Pc$ $\cross Pc$ is $ \xi_{2\delta}$ for $Pc$ realized as areal form \mathrm{h}$ $\nu$ to $Ao_{0}\cong^{u}1\mathrm{n}\mathrm{d}_{P_{\mathbb{C}}}^{G_{\mathbb{C}}}(\xi_{2\delta}\mathrm{H} 1)\cong u1\mathrm{n}\mathrm{d}_{p_{\mathbb{c}}}^{g_{\mathbb{c}}}(1\mathrm{h} \xi_{2\delta})$ Therefore Problem 12 reduced to problem of existence of intertwining operators $t\in \mathbb{c}$ For we define following generalized Verma module: $M_{\mathfrak{p}}(t\delta)=U(\mathrm{g})\otimes_{U(\mathfrak{p})}\xi_{t\delta}$ The following result is well-known Proposition 23 For $t_{1}$ $t_{2}\in 2\mathbb{Z}$ $u1\mathrm{n}\mathrm{d}_{p_{\mathbb{c}}}^{g_{\mathbb{c}}}(\xi_{t_{1}\delta}\mathrm{h} \xi_{i_{2}\delta})\cong(m_{\mathfrak{p}}(-t_{1}\delta)\mathrm{h} M_{\mathfrak{p}}(-t_{2}\delta))_{K_{\mathbb{C}}}^{*}$ So our Problem 12 is seriouly related to existence of homomorphisms between generalized Verma modules In fact following result is known

4 $u_{1\mathrm{n}\mathrm{d}_{p_{\mathbb{c}}}^{g_{\mathbb{c}}}(1\mathrm{h}}$ $1)$ $arrow$ $u\mathrm{i}\mathrm{n}\mathrm{d}_{p_{\mathbb{c}}}^{g_{\mathbb{c}}}(1\mathrm{h} \xi_{2\delta})$ corresponding such \mathrm{b}$ $\subseteq \mathfrak{p}$ 74 Theorem 24 ([Matumoto 1 gg3]) Let t be a non-negative even integer Then we have $M_{\mathfrak{p}}(-(t+2)\delta)\mapsto M_{\mathfrak{p}}(t\delta)$ and only wqwp is a Duflo involution in Weyl group for $(\mathrm{g} If wowp i$\mathrm{s}$ aduflo involution using Propostion 22 we have: $arrow d//$ $u1\mathrm{n}\mathrm{d}_{p_{\mathbb{c}}}^{g_{\mathbb{c}}}(\xi_{2\delta}\mathrm{h}\downarrow 1)$ $u1\mathrm{n}\mathrm{d}_{p_{\mathbb{c}}}^{g_{\mathbb{c}}}(\xi_{2\delta}\mathrm{h}\downarrow \xi_{2\delta})$ In fact we have : $Ao_{0}\mapsto Theorem 25 $(\mathrm{g} Weyl group for u1\mathrm{n}\mathrm{d}_{p_{\mathbb{c}}}^{g_{\mathbb{c}}}(\xi_{2\delta}\mathrm{h} \xi_{2\delta})$ and only $w_{0}w_{\mathfrak{p}}$ is a Duflo involution in \S 3 Type Acase As we seen in case of complex groups statement in Problem 12 is not correct in general However for type Agroups we have affirmative answers 31 $\mathrm{g}\mathrm{l}(n \mathbb{c})$ $\mathfrak{y}$ $\mathfrak{h}\subseteq We retain notation in \S 2 We fix aborel subalgebra that We denote $\Pi$ $\mathrm{b}$ $(\mathrm{g} by basis of root system with respect to to We denote by $S$ $\Pi$ $\mathfrak{p}$ subset of corresponding to Assumption 11 holds and only is compatible with $S$ $\mathrm{a}$ symmetry of Dynkin diagram For aweyl group of type each involution is a Duflo involution Hence we have: Theorem 36 Under Assumption 11 we have $A_{\mathcal{O}_{0}}\mathrm{e}arrow \mathrm{i}u\mathrm{n}\mathrm{d}_{p_{\mathbb{c}}}^{g_{\mathbb{c}}}$ $(\xi_{2\delta}\mathrm{h} \xi_{2\delta})$ 32 $\mathrm{g}\mathrm{l}(n \mathbb{r})$ $\mathrm{g}\mathrm{l}(n\mathbb{r})$ Speh proved any derived functor module of is parabolically induced from external tensor product of some s0-called Speh representations and possibly aone-dimensional representation Using this fact we can reduce Problem 12 to embedding Speh representations into degenerate principal series More pricisely we consider and let $G=\mathrm{G}\mathrm{L}(2n\mathbb{R})$ $P$ be amaximal parabolc $\mathrm{g}\mathrm{l}(n\mathbb{r})\cross \mathrm{g}\mathrm{l}(n \mathbb{r})$ subgroup whose Levi part is isomorphic to Then $X=G\mathbb{C}/P\mathbb{C}$ contains aunique open $G$ -orbit(say $O$ ) In this setting Assuption 11 holds The fine structure of degenerate principal series for $P$ has already been studied precisely $1999][\mathrm{B}\mathrm{a}\mathrm{r}\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{c}\mathrm{h}$ ([Sahi 1995] [Zhang 1995] [Howe-Lee -Sahi-Speh1988])From ir results we have: $A\mathrm{o}$ $\mapsto u_{1\mathrm{n}\mathrm{d}_{p}^{g}(\xi_{2\delta})}$ $Ao$ $\mathrm{c}arrow^{u}1\mathrm{n}\mathrm{d}_{p}^{g}(\xi 2\delta\otimes\omega)$ We can deduce an affirtive answer to Problem 12 from this is odd $n$ is even $n$

5 $\mathrm{g}\mathrm{l}(n \mathbb{h})$ In this case we also have an affirmative answer to Problem 12 The argument is similar to $\mathrm{g}\mathrm{l}(n \mathbb{r})$ (and easier than) case of 34 $\mathrm{u}(m$n) Let $G=\mathrm{U}(m n)$ and let $P$ be an arbitrary prarabolic subgroup of In this case Assumption $G$ $\mathcal{v}$ 11 automatically holds We denote by set of open -orbits on $G$ $X=Gq/Pq$ In fact we have: Socle $(^{u}1\mathrm{n}\mathrm{d}_{p}^{g}(\xi_{2\delta}))=\oplus Ao$ $0\in \mathcal{v}$ References [Barbasch-Sahi-Speh 1988] D Barbasch S Sahi B Speh Degenerate series representa $\mathrm{g}\mathrm{l}(2n \mathbb{r})$ tions for and Fourier analysis in :Symposia Mamatica Vol XXXI (Rome 1988) $\mathrm{g}\mathrm{l}_{n}(\mathbb{c})$ [Howe-Lee 1999] R Howe S T Lee Degenerate principal series representations of $\mathrm{g}\mathrm{l}_{n}(\mathbb{r})$ and J Fund Anal 166 (1999) [Johnson 1990] Degenerate principal series and compact groups Math Ann 287 (1990) [Lee 1994] S T Lee On some degenerate principal series representations of n) J $\mathrm{u}(n$ Fund Anal 126 (1994) [Matsuki 1988] T Matsuki Closssure relations flor orbits on affine symmetric sspaces under action of parabolic subgroups Intersections of associaated orbit Hiroshima Math J 18 (1988) [Matumoto 1993] H Matumoto On existence of homomorphisms between scalar generalized Verma modules in: Contemporary Mamatics Amer Math Soc Providence RI 1993 [Sahi 1993] S Sahi Unitary representations on Shilov boundary of asymmetric tube domain in: Representation Theory of Groups and Algebras Contemporary Math vol 145 American Mamatical Society [Sahi 1995] S Sahi Jordan algebras and degenerate principal series J Reine Angew Math 462 (1995) 1-18 [Speh $\mathrm{g}\mathrm{l}(n \mathbb{r})$ 1983] B Speh Unitary representations of with non-trivial (g K) cohomology Invent Math 71 (1983) [Vogan 1997] D A Vogan Jr Cohomology and group representations in Representation ory and automorphic forms (Edinburgh 1996) Proc Sympos Pure Math Amer Math Soc Providence RI 1997 [Vogan-Zuckerman 1984] D A Vogan Jr and G Zuckerman Unitary representations with non-zero cohomology Compositio Math 53 (1984) [Zhang 1995] G K Zhang Jordan algebras and generalized principal series representations Math Ann 302 (1995)