Dual Pairs in the Pin-Group and Duality for the Corresponding Spinorial Representation
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Dual Pairs in the Pin-Group and Duality for the Corresponding Spinorial Representation ´ ´ 1 · Gang Liu2 · Allan Merino3 Clement Guerin Received: 14 October 2019 / Accepted: 19 October 2020 / © Springer Nature B.V. 2020
Abstract In this paper, we give a complete picture of Howe correspondence for the setting (O(E, b), P in(E, b), ), where O(E, b) is a real orthogonal group, P in(E, b) is the twofold Pin-covering of O(E, b), and is the spinorial representation of P in(E, b). More precisely, for a dual pair (G, G ) in O(E, b), we determine explicitly the nature of its preim˜ G˜ ) is always a ˜ G˜ ) in P in(E, b), and prove that apart from some exceptions, (G, ages (G, dual pair in P in(E, b); then we establish the Howe correspondence for with respect to ˜ G˜ ). (G, Keywords Dual pairs · Pin group · Spinorial representation · Duality Mathematics Subject Classification (2010) 22E46 · 20G05
1 Introduction The first duality phenomenon has been discovered by H. Weyl who pointed out a correspondence between some irreducible finite dimensional representations of the general linear group GL(V ) and the symmetric group Sk where V is a finite dimensional vector space
Presented by: Michel Brion Allan Merino
[email protected] Cl´ement Gu´erin [email protected] Gang Liu [email protected] 1
CUFR de Mayotte, 8 rue de l’Universit´e - Iloni - BP 53, 97660 Dembeni, France
2
Institut Elie Cartan de Lorraine, Universit´e de Lorraine, 3 rue Augustin Fresnel, 57073 Metz, France
3
Department of Mathematics, National University of Singapore, Block S17, 10, Lower Kent Ridge Road, Singapore, 119076, Republic of Singapore
C. Gu´erin et al.
over C. Indeed, considering the joint action of GL(V ) and Sd on the space V ⊗d , we get the following decomposition: V ⊗d = λ ⊗ σ λ, )π (Vλ ,λ)∈GL(V
)π is the set of equivalence classes of irreducible finite dimensional repwhere GL(V resentations of GL(V ) such that H omGL(V ) (Vλ , V ⊗d ) = {0} and σλ is an irreducible representation of Sd . The key point in this duality is that the commutator in End(V ⊗d ) of the algebra generated by GL(V ) is precisely the algebra generated by Sd . Later, R. Howe [8] proved a similar phenomenon involving some particular subgroups of the symplectic group. Let (W, ·, ·) be a real symplectic space and (G, G ) be an irreducible ) reductive dual pair in the corresponding symplectic group Sp(W ). We denote by Sp(W the metaplectic group and by (ω, H ) the metaplectic representation. One can prove that ˜ and G ˜ of G and G respectively form a dual pair in the metaplectic group the preimages G ). We denote by R (G, ˜ ω) the set of infinitesimal classes of irreducible admissible Sp(W ˜ which can be realised as a quotient of H ∞ by a closed ω∞ (G)˜ representations of G invariant subspace. R. Howe proved that we have a one-to-one correspondence between ˜ ω) and R (G ˜ , ω), whose graph is R (G ˜G ˜ , ω). This correspondence is known as R (G, Howe correspondence or Theta correspondence. We can define in a similar way th
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