ON THE POSITIVITY OF KIRILLOV'S CHARACTER FORMULA EHSSAN KHANMOHAMMADI Abstract. We give a direct proof for the positivity of Kirillov's character on the convolution algebra of smooth, compactly supported functions on a connected, simply connected nilpotent Lie group G. Then we use this positivity result to construct a representation of G×G and establish a G×G-equivariant isometric isomorphism between our representation and the Hilbert{Schmidt operators on the underlying representation of G. Moreover, we show that the same methods apply to coadjoint orbits of other groups such as SL(2; R) under additional hypotheses that are automatically satisfied in the nilpotent case. Kirillov's character formula, Coadjoint orbit, Nilpotent Lie group, Positivity, Quantization, Polarization, The GNS construction. AMS MSC: 22E30, 22E27, 46L05. 1 1. Introduction 2 In his fundamental paper [12], Kirillov proved that coadjoint orbits of a con- 3 nected, simply connected nilpotent Lie group correspond, under quantization, to 4 the equivalence classes of its irreducible unitary representations. The theory of 5 geometric quantization due to Kirillov, Kostant [16], and Souriau [22] has shown 6 that this close connection extends to many other groups. 7 Kirillov's character formula says that the characters of irreducible unitary rep- 8 resentations of a Lie group G \should" be given by an equation of the form Z −1=2 i`(X)+σ (1.1) Tr πO(exp X) = j (X) e O ∗ 9 where O is the coadjoint orbit in g corresponding to πO 2 Gb, σ is a canonical 10 symplectic measure on O, and j is the analytic function on g defined by the formula sinh(ad X=2) j(X) = det : ad X=2 11 Kirillov proved his character formula for simply connected nilpotent and simply 12 connected compact Lie groups [12, 13] and conjectured its universality. The valid- 13 ity of this conjecture has been verified for some other classes of Lie groups, most 14 notably for the case of tempered representations of reductive Lie groups by Ross- 15 mann [20]. Moreover, Atiyah{Bott [2] and Berline{Vergne [3] following the work 16 of Duistermaat{Heckman [6], have shown that for compact Lie groups, Kirillov's 17 character formula is equivalent to the Weyl character formula. 18 Our main contributions in this paper are as follows. Rather than construct 19 representations, compute their characters, and compare the results with Kirillov's 20 character formula, we work directly with Kirillov's formula and give direct argu- 21 ments to prove positivity in the sense of operator algebras. More precisely, we give 1 2 EHSSAN KHANMOHAMMADI 22 a direct proof for the positivity of Kirillov's character on the convolution algebra 23 of the Schwartz functions on a connected nilpotent Lie group (Theorem 3.8). The 24 fact that Kirillov's character defines a positive trace is remarkable in light of the 25 Gelfand{Naimark{Segal (GNS) construction from operator algebra theory, since 26 GNS implies, roughly speaking, that any positive linear functional on a Lie group 27 is the character of a group representation. In the last section, we use our positivity 28 result and the GNS to construct a representation of G × G for a connected, simply 29 connected nilpotent Lie group G. Then in Theorem 5.3 we discuss the relation be- 30 tween the representations that we have obtained from the GNS construction, and 31 the Kirillov representations corresponding to the coadjoint orbits of G. Indeed, we 32 prove that there is a G × G-equivariant isometric isomorphism between our repre- 33 sentation and the Hilbert{Schmidt operators on the underlying representation of G. 34 We deal with nilpotent groups this way, thereby recovering some of the first results 35 obtained by Kirillov. As seen in Theorem 4.9, the same methods apply to coadjoint 36 orbits of other groups such as SL(2; R) in the presence of supplementary geomet- 37 ric hypotheses on the orbits that are automatically satisfied in the nilpotent case. 38 One hypothesis is the existence of a real polarization; a second is a completeness 39 hypothesis on the (locally affine) fibers of the associated Lagrangian fibration. 40 2. Some Notations and Background Material 41 In this section we fix some notations and collect several standard results that we 42 will use freely in the sequel. 43 Definition 2.1. Let G be a locally compact group G with a left Haar measure dµ. 44 The modular function ∆: G ! (0; 1) is defined via Z Z ∆(y) f(xy) dµ(x) = f(x) dµ(x) G G 1 45 for x and y in G and all f 2 L (G) so that Z Z f(x−1) dµ(x) = f(x)∆(x−1) dµ(x): G G −1 46 If G is a Lie group, then it is well known that ∆(g) = j det Ad(g )j. This 47 implies, for instance, that any connected nilpotent Lie group is unimodular, that 48 is, for such groups the modular function is constant and everywhere equal to one. Definition 2.2. Let G be a locally compact group with a left Haar measure dµ. Given functions f and g on G, their convolution f ∗ g is the function on G defined by Z f ∗ g(x) = f(y)g(y−1x) dµ(y) G Z = f(xy)g(y−1) dµ(y) G 49 whenever one (and hence both) of these integrals makes sense. 50 Let (π; Vπ) be a unitary representation of a locally compact group G. Then π 1 51 induces a continuous homomorphism of Banach ∗-algebras from L (G) to the space ON THE POSITIVITY OF KIRILLOV'S CHARACTER FORMULA 3 52 of bounded operators B(Vπ) via Bochner integration. By abuse of notation, we 53 denote this induced ∗-homomorphism by π. Hence, Z 1 π : L (G) !B(Vπ) and π(f) = f(g)π(g) dg: G 1 54 Here we are using the facts that L (G) is a Banach algebra under convolution, 1 55 which is just the usual group ring CG if G is finite, and that L (G) has a natural ∗ 56 isometric (conjugate-linear) involution f 7! f , where f ∗(x) = f(x−1)∆(x−1): 57 Note that these definitions are formulated so that f ∗g = L(f)(g) for the left regular 1 58 representation (L; L (G)). 59 Definition 2.3. We say that an irreducible unitary representation π of a Lie group 60 G has a global or Harish-Chandra character Θ if π(f) is of trace class for all f 2 1 61 Cc (G) and moreover f 7! Θ(f) = Tr π(f) is a distribution. 62 The next two results provide sufficient conditions for π(f) to be trace class. 63 Theorem 2.4 (Harish-Chandra [8]). Let G be a connected semisimple Lie group 64 with finite center. Then for every irreducible unitary representation π of G and 1 65 every f 2 Cc (G), the operator π(f) is trace class and the map f 7! Tr π(f) is a 66 distribution on G. 67 Theorem 2.5 (Kirillov [12]). Let G be a nilpotent Lie group. Then for every 68 irreducible unitary representation π of G and every Schwartz function f 2 S(G), 69 the operator f 7! Tr π(f) is trace class and the map f 7! Tr π(f) is a tempered 70 distribution on G. 71 Let (π; Vπ) be an irreducible unitary representation. Observe that the above ∗ ∗ 72 discussions imply that π(f ∗ f ) = π(f)π(f) . Thus, the distributional character 73 of (π; Vπ), whenever defined, is convolution-positive (or positive for short) in the 74 sense that ∗ ∗ 2 (2.1) Tr π(f ∗ f ) = Tr π(f)π(f) = kπ(f)kHS ≥ 0: 75 To close this section, we introduce Kirillov's character formula which says that the 76 characters χ of irreducible unitary representations of a Lie group G \should" be 77 given by an equation of the form Z −1=2 i`(X) (2.2) χ(exp X) = j (X) e dµO(`): O ∗ 78 Here O is a coadjoint orbit in g , µO is the canonical (or Liouville) symplectic 79 measure on O, and j is the Jacobian of the exponential map exp: g ! G given by 80 the formula sinh(ad X=2) j(X) = det ad X=2 81 whenever G is unimodular. This character formula should be interpreted as an 82 equation of distributions on a certain space of test functions on g as follows. For 83 all smooth functions f compactly supported in a sufficiently small neighborhood of 84 the origin in g, Z Z Z i`(X) −1=2 (2.3) Tr f(X)π(exp X) dX = e f(X)j (X) dXdµO(`) g O g 4 EHSSAN KHANMOHAMMADI 85 where π is the representation with character χ. 86 Theorem 2.6 (Kirillov [12]). Suppose G is a connected nilpotent Lie group. Then 87 the irreducible unitary representations of G are in natural one-to-one correspon- 88 dence with the integral orbits of G. Moreover, if G is simply connected and π is an 89 irreducible unitary representation of G with the associated coadjoint orbit O, then 90 (2.3) holds. 91 3. Nilpotent Lie Groups 92 Nilpotent Lie groups and their representations have been studied extensively in 93 the literature. From the many papers by Corwin, Greanleaf, Lipsman, Pukanzsky 94 and others we only cite [15], [18], and [21] and refer the reader to [4] for a more 95 comprehensive list of bibliographies.