Heisenberg groups over finite fields
نویسنده
چکیده
ing this computation, for given k-vectorspace V with non-degenerate alternating form 〈, 〉, put a Lie algebra [2] structure h on V ⊕ k by Lie bracket [v ⊕ z, v′ ⊕ z′] = 0⊕ 〈v, v′〉 In exponential coordinates on H, the exponential map h→ H with H ≈ V ⊕ k is notated exp(v ⊕ z) = v ⊕ z with Lie group structure on H by (v ⊕ z) · (v′ ⊕ z′) = (v + v′)⊕ (z + z′ + 〈v, v ′〉 2 ) (exponential coordinates in H) In the Lie algebra/exponential coordinates, any k-linear map g : V → V preserving the alternating form gives an automorphism τg of the Lie algebra h ≈ V ⊕ k and and of the Lie group H ≈ V ⊕ k, by τg(v ⊕ z) = gv ⊕ z (same expression for both h and H) 4. Segal-Shale-Weil/oscillator representations The uniqueness of the representation π with fixed non-trivial central character ω of the Heisenberg group H = V ⊕ k, almost gives a representation of the isometry group Sp(V ) = Sp(V, 〈, 〉) of 〈, 〉 on V . What literally arises is a projective representation ρ of Sp(V ) on π, meaning that each ρ(g) is ambiguous by a scalar depending on g, as follows. Let π be an irreducible of H on a complex vectorspace X, with non-trivial central character ω. The twist π given by π(h)(x) = π(τgh)(x) is a representation of H on the same vectorspace X. By uniqueness, the twist π is H-isomorphic to π. [3] By Schur’s lemma, this isomorphism is unique up to constants. That is, there is an H-isomorphism ρ(g) : (π, X)→ (π,X), unique up to scalar multiples. [2] In positive characteristic, generally a Lie algebra needs some further structure to behave properly, but in this simple situation the potential troubles do not appear. [3] The map (π,X)→ (π, X) by the identity mapping on X is most likely not an H-homomorphism.
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