# Primitive Ideals and Unitarity

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1 Primitive Ideals and Unitarity Dan Barbasch June

2 The Unitary Dual NOTATION. G is the rational points over F = R or a p-adic field, of a linear connected reductive group. A representation (π, H) on a Hilbert space is called unitary, if H admits a G invariant positive definite inner product. PROBLEM. Classify the unitary dual of G. ALGEBRAIC PROBLEM. By results of Harish-Chandra, and others, this is turned into an algebraic problem, about (g, K) modules. In this talk, we mostly deal with complex reductive group viewed as a real group. The answer is known for classical groups, but not for groups of exceptional type. For Spin(n, C), the complete answer is also not known. 2

3 Admissible Modules Notation - B = HN is a Borel subgroup, with H = T A, T the toral part, A the split part. The corresponding Lie algebras are b, h, a, n. - g := Lie(G), g c = g L g R, U(g c ) = U(g L ) U(g R ). - the positive roots corresponding to b, Π the simple roots, W the Weyl group. - K a maximal compact subgroup satisfying K B = T, and G = K B. - µ a weight of T, V (µ) the irreducible representation of K with extremal weight µ. - X(µ, ν) the principal series representation Ind G B [C µ C ν ] K finite, where ν Â. Let L(µ, ν) be the unique irreducible subquotient of X(µ, ν) containing the K-type V (µ). 2-1

4 Theorem (P-R-V, Zh). Any irreducible (g c, K) module is isomorphic to an L(µ, ν). L(µ, ν) = L(µ, ν ) if and only if there is w W such that wµ = µ, wν = ν. We will parametrize representations by (λ L, λ R ) such that µ = λ L λ R, ν = λ L + λ R. Hermitian Structure. U(g c ) has a operation which interchanges g L and g R. A module L(µ, ν) is hermitian if and only if there is w W such that wµ = µ, wν = ν. Annihilator. The annihilator of an L(λ L, λ R ) is of the form U(g) Ǐ + I U(g), with I a primitive ideal (according to Duflo, the annihilattor of an irreducible highest weight module). When the module is hermitian, I = Ǐ. In such a case, we write λ for the infinitesimal character. We can think of it as the W -orbit of λ L, or equivalently λ R. 2-2

5 Arthur Parameters Let ǧ be the dual Lie algebra. According to Arthur s conjectures, attached to each (Ǧ conjugacy class of) Ψ : W R SL(2) ǧ, there should be a packet of representations which are unitary. Their characters should have nice properties with respect to endoscopic groups. The restriction of Ψ to SL(2) determines a nilpotent orbit in the centralizer of Ψ(W R ), which is a Levi component, say M Ψ. Ψ(W R ) is assumed to be bounded. Ψ WR determines a unitary character ψ of M Ψ. On the other hand Ψ SL(2) also determines an Arthur parameter for M Ψ. The packet for Ψ is obtained by tensoring the representations associated to Ψ SL(2) with ψ, and inducing up to G. 3

6 Let h := dψ( 1 0 ), 0 1 e := dψ( 0 1 ), 0 0 f := dψ( 0 0 ), 1 0 Definition (B-V). The Arthur parameter is called special unipotent, if Ψ WR is trivial. The packet is defined to be the set of irreducible representations with infinitesimal character λ := 1 2 h, and annihilator the unique maximal primitive ideal I with this infinitesimal character. Since such maps Ψ are in one-to-one correspondence with nilpotent orbits, we use Ǒ as parameters. It is sufficient to consider the case when Ǒ is even; this means λ Ǒ is integral. If not, there is a reduction to this via unitary induction. 4

7 Character Formulas of Unipotent Parameters - - Unip(Ǒ) the set of unipotent represntations attached to Ǒ. G(Ǒ) the Grothendieck group spanned by the characters of the Unip(Ǒ). When Ǒ is special, Lusztig defines a finite group A(Ǒ) with the following properties: - A(Ǒ) is a quotient of A(Ǒ), the component group of the centralizer of an element in Ǒ. - There is a bijection [x] [A(Ǒ)] σ x, between conjugacy classes in left cell attached to I(Ǒ). A(Ǒ) and representations in the 5

8 - There is a bijection χ Â(Ǒ) L χ Unip(Ǒ). - There are virtual characters R σx (which are computable in terms of X(λ L, λ R )) so that the change of bases between the R x and L χ is the same as between the class functions χ x, and characters χ of A(Ǒ). These results can be found in [B-V]. Lusztig s results on the character theory of the finite Chevalley groups, and their consequences for primitive ideals play a prominent role. These formulas imply the unitarity of the special unipotent representations in the classical cases. 6

9 Unitary Dual The general strategy to classify the unitary dual is to find a finite set of representations which we will call basic (with nice properties) so that the full dual is obtained from them by unitary induction and complementary series. To make this idea more precise, we need the notion of induced nilpotent, of Lusztig-Spaltenstein. Let p = m + n g be a parabolic subalgebra, and O m a nilpotent orbit in m. Then there is a unique nilpotent orbit O g with the property that its intersection with O m + n is a dense subset. We write O = Ind g m[o m ]. The orbit O only depends on O m m, not on the particular parabolic subalgebra. A representation π has a wave front set denoted W F (π), which in the case of complex groups is the closure 7

10 of a single nilpotent orbit. If π = Ind G M [π M ], then W F (π) = Ind g m[w F (π M )]. So the basic set should consist of representations of G with WF-set cuspidal, i.e. not induced from any proper Levi component. It is natural to expect that this set should consist of special unipotent representations. There aren t enough of them to describe the unitary dual in this way. For example the oscillator/metaplectic/segal-shale-weil representation must be one of the basic representations, but is not special unipotent. 8

11 Basic Representations For each classical Lie algebra, we give a set of nilpotent orbits. For each such orbit we describe an infinitesimal character λ(o). The basic representations U nip(o) are the ones with this infinitesimal character, and annihilator the maximal primitive ideal. The WF-set is the closure of O. - Unip(O) is in one-to-one correspondence with Â(O). - L 1 has K-structure equivalent to R(O), the ring of regular functions on O. - Character formulas similar to the ones for special unipotent representions hold. - The K-spectrum of L χ has a structure of S(g) module, it is essentially dim χ times R(O). 9

12 Remark This parametrization is along the lines of the orbit method. The combinatorics defining λ(o) are in the spirit of Lusztig s symbols for primitive ideal cells. The list of nilpotents is larger than all cuspidal ones. Also some special unipotent parameters are missing, for example the orbit with Jordan blocks (22) in Sp(2) appears as WF-set for special unipotent parameters (1, 0) as well as (1/2, 1/2). Only the first one is listed as basic. These two parameters are part of a complementary series. 10

13 Type B Let (c 0 )(c 1 c 2 )... (c 2m c 2m+1 ) c 2i > c 2i+1 be the sizes of the columns of the Jordan blocks of O in decreasing order. Then the coordinates of λ(o) are the union of the coordinates (c 0 ) ( c 0 1,..., 1/2) 2 (c 2i 1 c 2i ) ( c (1) 2i 1 2,..., c 2i 2 ) 11

14 Type C Let (c 0 c 1 )... (c 2m 1 c 2m )(c 2m+1 ) c 2i 1 > c 2i be the sizes of the columns of the Jordan blocks of O. Then the coordinates of λ(o) are the union of the coordinates (c 2m+1 ) ( c 2m+1 1,..., 0) 2 (c 2i c 2i+1 ) ( c (2) 2i 2,..., c 2i+1 2 ) 12

15 Type D Let (c 0 c 2m 1 )(c 1 c 2 )... (c 2m 3 c 2m 2 ) c 2i > c 2i+1 be the sizes of the columns of the Jordan blocks of O. Then the coordinates of λ(o) are the union of the coordinates (c 0 c 2m 1 ) ( c 0 2 1,..., c 2m 1 ) 2 (c 2i 1 c 2i ) ( c (3) 2i 1 2,..., c 2i 2 ) 13

16 Primitive Ideals and Unitarity Lie Algebras of Exceptional Type We will concentrate on nilpotent orbits related to what Lusztig calls triangular nilpotent orbits. T ype Label Component Group G 2 G 2 (a 1 ) S 3 F 4 F 4 (a 3 ) S 4 E 6 D 4 (a 1 ) S 3 E 8 E 8 (a 7 ) S 5 14

17 Primitive Ideals and Unitarity These are interesting nilpotents in many ways. In particular the component groups A(O) are not powers of Z 2. These nilpotent orbits are even. To each of them corresponds an Arthur parameter, which is h/2, but computed in terms of the dual roots. The Arthur packets, and their character theory are computed in [BV]. In this talk I am interested in unitary parameters for a set of nilpotent orbits which are related to the triangular ones. They are not special unipotent. The next slides will describe them in detail. 15

18 Primitive Ideals and Unitarity G 2 (a 1 ) Lusztig s matching of Weyl group representations is as follows (Carter s notation): Conjugacy Class W eyl representation Springer correpondence (1) (2, 1) G 2 (a 1 ) (g 2 ) (1, 3) A l 1 (g 3 ) (2, 2) A s 1. A l 1 and A s 1 are the nilpotent orbits for which we want to find unitary parameters. The label A l 1 indicates that one should consider the unitarily induced module from the trivial representation on the Levi component of this type. The WF-set is G 2 (a 1 ). The induced representation is irreducible, and we can form a complementary series. In terms of the realization of the root 16

19 Primitive Ideals and Unitarity system in Bourbaki, the parameter is (1/2, 1, 1/2) + η(1, 0, 1). The parameter is the special unipotent one for G 2 (a 1 ). The representation is reducible, there are two factors which are unipotent. Being an endpoint of a complementary series, the two factors are unitary. The next reducibility point is at η = 5/6. The WF-set of the spherical representation is A l 1. The integral system of this infinitesimal character is A 2. This is the right representation, provided it is unitary. The answer is yes, the two factors do not change sign. 17

20 Primitive Ideals and Unitarity In the case of A s 1, the situation is simpler. The parameter is (0, 1/2, 1/2) + ν(2, 1 1). It is special unipotent at ν = 1/2, but the induced module is irreducible. The next reducibility gives a parameter integral for A 1 A 1, the spherical representation is unitary and has WF-set A s 1. In fact for G 2, every nilpotent orbit has a family (some several) of unitary representations with the desired properties. 18

21 Primitive Ideals and Unitarity D 4 (a 1 ) Again, in Carter s notation, Conjugacy Class W eyl representation Springer correpondence (1) (80, 7) D 4 (a 2 ) (g 2 ) (60, 8) A 3 A 1 (g 3 ) (10, 9) 2A 2 A 1. As for G 2, the induced from the trivial representation on A 3 A 1 has WF-set D 4 (a 1 ), and a complementary series; the parameter is ( 5/4, 1/4, 3/4, 5/4, 1/4, 3/4, 3/4, 3/4) + ν(1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2). (4) 19

22 Primitive Ideals and Unitarity The special unipotent parameter for D 4 (a 1 ) is a ν = 1/2. There are two factors, but the signature does not change. So at the next reducibility point, ν = 1, there is a spherical unitary representation with WF-set A 3 A 1. The induced from the trivial representation on 2A 2 A 1 has similar properties. The special unipotent parameter for D 4 (a 1 ), arises at ν = 1/2. The representation is irreducible. The next reducibility point is at ν = 2/3. The spherical irreducible module is unitary, and has WF-set 2A 2 2A 1. 20

23 Primitive Ideals and Unitarity F 4 (a 3 ) Conjugacy Class W eyl Springer U nitary (1) (12, 4) F 4 (a 3 ) (2, 1, 1, 0) (g 2) (9, 6) B 2 (2, 2, 1, 0) (g 4 ) (4, 7) A 2 Ã 1 (11/4, 5/4, 1/4, 3/4) (g 3 ) (6, 6) Ã 2 A 1 (8/3, 1/3, 1/3, 1) (g 4 ) (16, 5) C 3 (a 1 ) (5/2, 1/2, 1, 0). For each of these nilpotent orbits, there is an induced nilpotent, and a complementary series with the same properties as in the previous cases. 21

24 Primitive Ideals and Unitarity Remark The nilpotent orbits B 2 and C 3 (a 1 ) are also unitarily induced, one from the cuspidal orbit 21 4 in C 3, the other from the orbit in B 3. The unitarily induced parameters are (0, 5/2, 3/2, 1/2) and (3/2, 3/2, 3/2,, 1/2). These are different from the endpoints of complementary series above. 22

25 Primitive Ideals and Unitarity Bibliography References [BV] D. Barbasch, D. Vogan Unipotent representations of complex semisimple groups Ann. of Math., 121, (1985), [V] D. Vogan The unitary dual of GL(n) over an archimedean field, Inv. Math., 83 (1986),

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