Local theta correspondence of depth zero representations and theta dichotomy.
Let \(F\) be a \(p\)-adic field with odd residual characteristic, with ring of integers \(\mathbb Z_F\) and let \(D\) be a central division algebra over \(F\) with an involution \(\tau\) and ring of integers \(\mathbb Z_D\). Let \(\varpi_F\) (resp. \(\varpi_D\)) be a uniformizer of \(F\) (resp. \(D\)), and let \(\psi\) be a continuous character of \(F\). Let \(\varepsilon\), \(\varepsilon'\) in \(\{-1,1\}\) with \(\varepsilon\varepsilon'=-1\), and let \((\mathcal V,\langle\;,\;\rangle)\) (resp. \((\mathcal V',\langle\;,\;\rangle')\)) denote a finite-dimensional vector space over \(D\), equipped with a nondegenerate \(\varepsilon\)-Hermitian form. One can define a skew-symmetric \(F\)-bilinear form \(\langle\!\langle\;,\;\rangle\!\rangle\) on the space \(\mathcal W=\mathcal V\otimes \mathcal V'\) by \(\langle\!\langle\;,\;\rangle\!\rangle={\text{Trd}}_{D| F}(\langle\;,\;\rangle\otimes \tau\circ\langle\;,\;\rangle')\) where \({\text{Trd}}_{D| F}\) denotes the reduced trace from \(D\) to \(F\), and the pair \((U(\mathcal V),U(\mathcal V'))\) forms a reductive dual pair over \(F\) in the symplectic group \({\text{Sp}}(\mathcal W)\). Let \(\widetilde{\text{Sp}} (\mathcal W)\) be the metaplectic cover of \({\text{Sp}}(\mathcal W)\), and let \(\widetilde U(\mathcal V)\) (resp. \(\widetilde U(\mathcal V'))\) denote the inverse image of \(U(\mathcal V)\) (resp. \(U(\mathcal V')\)) in \(\widetilde {\text{Sp}}(\mathcal W)\). Since \(\widetilde U(\mathcal V)\) and \(\widetilde U(\mathcal V')\) commute with each other in \(\widetilde {\text{Sp}}(\mathcal W)\), their product \(\widetilde U(\mathcal V)\cdot \widetilde U(\mathcal V')\) is a subgroup of \(\widetilde {\text{Sp}}(\mathcal W)\), and, by restricting the Weil representation \(\omega_\psi\) of \(\widetilde{\text{Sp}}(\mathcal W)\) to that subgroup, one obtains a one-to-one correspondence (the Howe correspondence or local theta correspondence) between irreducible smooth representations of \(\widetilde U(\mathcal V)\) and irreducible smooth representations of \(\widetilde U(\mathcal V')\). The explicit determination of the correspondence is extremely difficult. In fact, very little is known except for few special cases. Let \(\lambda_F\) denote the conductoral exponent of \(\psi\), that is, the smallest integer \(n\) such that the restriction of \(\psi\) to \(\varpi^n\mathbb Z_F\) is trivial. Define \(\lambda:=\lambda_F\) if \(D\) is \(F\) or an unramified quadratic extension of \(F\), and \(\lambda:=2\lambda_F-1\) otherwise. Let \(\kappa\), \(\kappa'\) be two integers such that \(\kappa+\kappa'=\lambda\). Let \(L\) be a {good lattice} in \(\mathcal V\), that is a lattice such that \(L^*\varpi\subset L\subset L^*\), where \(L^*=\{v\in \mathcal V: \langle v,l\rangle\in\varpi^\kappa\mathbb Z_D\}\). Let \(G_L\) be the stabilizer of \(L\) in \(G=U(\mathcal V)\) and define \(G_{L,0^+}:= \{g\in G: (g-1)L^*\subset L\), \((g-1)L\subset L^*\varpi_D\}\) (a normal subgroup of \(G_L\)). Fix an Iwahori subgroup \(I\) of \(G\) and a splitting \(\beta_I: I\to\widetilde I\). An irreducible smooth representation \((\pi,V)\) of \(\widetilde G=\widetilde U(V)\) is said to be of level zero (or depth zero) if \(V^{\beta_I(G_{L,0^+})}\) (the vectors fixed by \(G_{L,0^+}\)) is some nonzero good lattice \(L\) such that \(G_{L,0^+}\subset I\). The following is the first main result of the paper: Let \((\pi,V)\), \((\pi',V')\) be irreducible smooth representations of \(\widetilde U(\mathcal V)\), \(\widetilde U(\mathcal V')\) respectively, which are paired by the Howe correspondence. Then \(\pi\) is of level zero if and only if \(\pi'\) is. The author calls a depth zero minimal \(K\)-type for \(G\) a pair \((G_L,\zeta)\) where \(\zeta\) is an irreducible representation of the finite reductive quotient \(\mathbb G_L=G_L/G_{L,0^+}\). One says that an irreducible smooth representation \((\pi,V)\) of \(G\) contains a minimal \(K\)-type \((G_L,\zeta)\) if \(V^{G_{L,0^+}}\) is nonzero and \(V^{G_{L,0^+}}\), as a representation of \(\mathbb G_L\), contains \(\zeta\). The concept of reductive dual pair can also be defined over a finite field. In particular, the Howe correspondence for a finite reductive dual pair is also defined although this correspondence is not one-to-one in general. Moreover, if \((G,G')=(U(\mathcal V),U(\mathcal V'))\) is a reductive dual pair over \(F\) and \(L\), \(L'\) are good lattices in \(\mathcal V\), \(\mathcal V'\) respectively, then \((\mathbb G_L,\mathbb G_{L'})\) is a reductive dual pair over the residue field of \(F\). The following is the second main result of the paper: Let \((G,G')=(U(\mathcal V),U(\mathcal V'))\) be a reductive dual pair. Let \((\pi,V)\), \((\pi',V')\) be irreducible smooth representations of \(\widetilde U(\mathcal V)\), \(\widetilde U(\mathcal V')\) respectively which are paired by the Howe correspondence. If \(\pi\) contains a depth zero minimal \(K\)-type \((G_L,\zeta)\) for some good lattice \(L\) such that \(G_{L,0^+}\subset I\), then \(\pi'\) contains a depth zero minimal \(K\)-type \((G_{L'},\zeta')\) such that \(\zeta\) and \(\zeta'\) are paired by the Howe correspondence for the dual pair \((\mathbb G_L,\mathbb G_{L'}')\). A reductive dual pair \((U(\mathcal V),U(\mathcal V'))\) is called split if two splittings \(U(\mathcal V)\to\widetilde U(\mathcal V)\) and \(U(\mathcal V')\to\widetilde U(\mathcal V')\) exist. If \(D\) is commutative, the author has defined in another paper a splitting \(\tilde\beta^L: U(\mathcal V)\to\widetilde U(\mathcal V)\) depending on a good lattice \(L\) in \(\mathcal V\). The advantage of the splitting \(\widetilde\beta_L\) is that \(\pi\) is of level zero if and only if \(\pi\circ\widetilde\beta_L\) is of level zero. Using the splitting \(\tilde\beta_L\) the author is able to describe the Howe correspondence of level zero supercuspidal representations for \(p\)-adic reductive dual pairs completely in terms of the Howe correspondence for finite reductive dual pairs as follows: Let \((G,G')=(U(\mathcal V),U(\mathcal V'))\) be a split reductive dual pair such that \(D\) is commutative. (i) Let \((\pi,V)\), \((\pi',V')\) be irreducible smooth representations of \(\widetilde U(\mathcal V)\), \(\widetilde U(\mathcal V')\) respectively. Assume that \(\pi\) contains a depth zero minimal \(K\)-type \((G_L,\zeta)\) for some good lattice \(L\) in \(\mathcal V\), that \(\pi\circ\widetilde\beta\) is a supercuspidal representation of \(U(\mathcal V)\) for some splitting \(\widetilde\beta:U(\mathcal V)\to\widetilde U(\mathcal V)\) and that \(\pi\otimes\pi'\) is a first occurrence in the Howe correspondence. Then \(\pi'\) contains a depth zero minimal \(K\)-type \((G_{L'},\zeta')\) such that \(\zeta\otimes\zeta'\) is a first occurrence in the Howe correspondence for the reductive dual pair \((\mathbb G_L,\mathbb G_{L'}')\) and both \(\zeta\), \(\zeta'\) are cuspidal representations. (ii) Suppose that \(\zeta\otimes\zeta'\) is a first occurrence in the Howe correspondence for the reductive dual pair \((\mathbb G_L,\mathbb G_{L'}')\) where both \(\zeta\), \(\zeta'\) are cuspidal. Then \(\pi\otimes\pi'\) is a first occurrence for the reductive dual pair \((G,G')\) where \(\pi\) is the irreducible smooth representation of \(\widetilde G\) such that \(\pi\circ\widetilde\beta^L\simeq\) c-Ind\(_{G_L}^G(\xi_L\otimes\zeta)\), \(\pi'\) is the irreducible smooth representation of \(\widetilde G'\) such that \(\pi'\circ\widetilde\beta^{L'}\simeq\) c-Ind\(_{G'_{L'}}^G(\xi_{L'}\otimes\zeta')\), and \(\xi_L\) (resp. \(\xi_{L'}\)) is a certain character of \(G_L\) (resp. \(G'_{L'}\)). Two interesting results about theta dichotomy are also proved in the paper.
scientific article; zbMATH DE number 1943286