LOCAL THETA CORRESPONDENCE AND

UNREFINED MINIMAL K-TYPES*

Shu-Yen Pan

0. Introduction 0.1. Unrefined minimal K-types of irreducible admissble representations of p-adic reductive groups are defined by A. Moy and G. Prasad [MP1, MP2]. In particular, a non-negative rational number can be associated to an irreducible admissble representation of a p-adic reductive group. This rational number is called the depth of the representation.

0.2. Let (G_{1}, G_{2}) be a type I (irreducible) reductive dual pair in a symplectic group Sp_{2n}(F). Let ψ be a fixed non-trivial additive character of F. Let ω_{ψ} be the Weil representation associated to ψ of the metaplectic cover S̃p_{2n}(F) of Sp_{2n}(F). By restricting ω_{ψ} to G̃_{1} · G̃_{2}, where G̃_{i} is the inverse image of G_{i} under the map S̃p_{2n}(F) → Sp_{2n}(F), we obtain a one-to-one correspondence between some irreducible admissible representations of G̃_{1} and irreducible admissible representations of G̃_{2}. The correspondence is called the local theta correspondence.

0.3. Let I be an Iwahori subgroup of a classical group G which is a member of some reductive dual pair. It is known the map Ĩ → I splits. By fixing an embedding I → G̃, the concepts of unrefined minimal K-types and depths can be generalized to irreducible admissible representations of G̃. We prove that the depths of the representations paired by the local theta correspondence for any (type I) reductive dual pair are equal.

1. Moy-Prasad filtrations 1.1. Basic constructions. Let F be a p-adic local field, O the ring of integers, ϖ a prime element, 𝔭 the maximal ideal, f the residue field of F. Let G be the group of F -rational points of a classical group over F. Given a point x in the Bruhat-Tits building of G, A. Moy and G. Prasad [MP1, MP2] define a decreasing filtration subgroups {G_{x,r}}_{r∈R^{+}∪{0}} of the parahoric subgroup G_{x} associated to x (i.e., G_{x} ⊇ G_{x,r} ⊇ G_{x,s} if 0 ≤ r ≤ s). Let G_{x,r^{+}} = ⋃_{s>r} G_{x,s}. It is known that G_{x,r} and G_{x,r^{+}} are normal subgroups of G_{x}, and G_{x,r}/G_{x,r^{+}} is a finite reductive group when r = 0 and it is a finite abelian group if r > 0. We can associate to x a sequence of non-negative real numbers {r_{i}}_{i≥0} such that r_{0} = 0 and r_{i} satisfies the conditions that G_{x,r_{i}} ≠ G_{x,r_{i}^{+}} and G_{x,r_{i}^{+}} = G_{x,r_{i+1}}.

1.2. filtration of a Lie algebra and its dual space. The similar filtration 𝔤_{x,r} of Lie algebra 𝔤 of G and the filtration 𝔤_{x,r}^{∗} of the dual space 𝔤^{∗} are also defined for x in the building and r any real number. We normalize the discrete valuation of F such that ϖ𝔤_{x,r} = 𝔤_{x,r+1}, ϖ𝔤_{x,r}^{∗} = 𝔤_{x,r+1}^{∗}. There is a natural isomorphism between the Pontrjagin dual (G_{x,r_{i}}/G_{x,r_{i+1}})^{∧} and the quotient 𝔤_{x,−r_{i}}^{∗}/𝔤_{x,−r_{i−1}}^{∗}.

* This is a note of the talk given at AMS summer research conference, Seattle, July 1997.

1.3. Optimal points. A certain finite set of points in a (closed) chamber, called a set of optimal points, satisfying some optimal properties is defined by Moy and Prasad. We know that for SL_{n}(F) (or GL_{n}(F)) the optimal points can be chosen to be the barycenters of symplices which are facets of a chamber.

1.3.1. Generalized barycenters. Define δ to be the smallest positive rational number such that ϕ + tδ is an affine root for any affine root ϕ and any integer t. Let Δ_{af} = {α_{0}, …, α_{n}} be a system of simple affine roots, identified with the vertices of the local Dynkin diagram of G. There are unique integers c_{i} such that ∑_{i} c_{i}α_{i} = δ. A vertex α_{i} is called terminal if there is at most one other vertex jointed to α_{i}. A vertex α_{i} is called exceptional if it is terminal and c_{i} = 2. A pair of vertices {α_{i}, α_{j}} is called an exceptional pair if α_{i} ≠ α_{j} and α_{i} and α_{j} form an orbit under the action of G on the local Dynkin diagram. Let Ξ be a non-empty subset of Δ_{af}, then Ξ determines a subsimplex C_{Ξ} of a chamber. Assign weights w_{i} to α_{i} ∈ Ξ in the following way: if {α_{i}, α_{j}} be an exceptional pair such that both α_{i}, α_{j} are in Ξ, (w_{i}, w_{j}) = (1, 1), (1, 2), or (2, 1); if α_{i} is an exceptional vertex, w_{i} = 1 or 2; if α_{i} is not an exceptional vertex and there is no α_{j} such that {α_{i}, α_{j}} is an exceptional pair contained in Ξ, w_{i} = 2. Then the weighted barycenter of the points α_{i} ∈ Ξ with weights w_{i} is called a generalized barycenter of C_{Ξ}.

1.3.2. There are at most 9 generalized barycenters of any C_{Ξ}. Many groups such as special linear groups, symplectic groups do not have any exceptional vertices or exceptional pairs, so all generalized barycenters are indeed (genuine) barycenters. For these groups there is only one (generalized) barycenter of C_{Ξ}.

1.4. (Unrefined) minimal K-types. An irreducible representation χ of G_{x,r_{i}}/G_{x,r_{i}^{+}} is called a non-degenerate representation if:

(i) χ is a cuspidal representation of G_{x,r_{i}}/G_{x,r_{i}^{+}} when r_{i} = 0; (ii) χ, realized as a residue class in 𝔤_{x,−r_{i}}^{∗}/𝔤_{x,−r_{i}^{+}}^{∗}, contains no nilpotent elements when

r_{i} > 0.

An (unrefined) minimal K-type of an irreducible admissible representation (π, V) of G is a pair (G_{x,r}, χ), where χ is a non-degenerate representation of G_{x,r}/G_{x,r^{+}}, and V^{G_{x,r^{+}}} , the space fixed by G_{x,r^{+}}, is non-trivial and, as a representation of G_{x,r}/G_{x,r^{+}}, contains χ.

1.5. Theorem (Moy-Prasad). Let (π, V) be an irreducible admissible representation of a p-adic reductive group G. Then there is a unique non-negative rational number ρ such that ρ is the smallest number r such that there exists an optimal point y with V^{G_{y,r^{+}}} ≠ {0}. Moreover, if ρ > 0 and y is a point such that V^{G_{y,ρ^{+}}} ≠ {0}, then the space V^{G_{y,ρ^{+}}} decomposes into non-degenerate representations of G_{y,ρ}/G_{y,ρ^{+}}. Conversely, if r > 0 and V^{G_{x,r^{+}}} contains a non-degenerate representation of G_{x,r}/G_{x,r^{+}} for some x, then r is equal to ρ.

This unique rational number ρ is called the depth of the representation (π, V).

1.5.1. Locations of optimal points. Let Q be the set of generalized barycenters in a fixed chamber of the building of a classical group G. The first result in [PY] is the following.

Theorem. Suppose that the local Dynkin diagram of a classical group G does not have any ramified ends, then the point y in Theorem 1.5 can be chosen from Q.

1.6. Good lattices. From now on, we shall assume that D is a divison algebra over F with an involution τ. Let O_{D} be the local ring in D, ϖ_{D} be a prime element. Let V be a right space over D. We assume that V is equipped with a non-degenerate ϵ-hermitian form ⟨, ⟩, where ϵ is 1 or −1. Let G be the group of isometries U(V) of (V, ⟨, ⟩). If L is a lattice in V, define

L^{∗} = { v ∈ V | ⟨v, l ⟩ ∈ O_{D} for all l ∈ L }.

L is said to be self-dual if L = L^{∗}, and is said to be almost-self-dual if L^{∗} = Lϖ_{D}^{t} for some integer t. A lattice L is called a good lattice if L^{∗}ϖ_{D} ⊆ L ⊆ L^{∗}.

1.6.1. Write V as an orthogonal direct sum V^{∘} ⊕ V_{1}, where V^{∘} is anisotropic, and V_{1} is a direct sum of hyperbolic planes. Then there is a unique good lattice A_{0} in V^{∘}. Choose a complete polarization V_{1} = X + Y, a basis x_{1}, …, x_{n} of X, the dual basis y_{1}, …, y_{n} of Y. Then

N_{i} = x_{1}O_{D}ϖ_{D} + ⋯ + x_{i}O_{D}ϖ_{D} + x_{i+1}O_{D} + ⋯ + x_{n}O_{D} + y_{n}O_{D} + ⋯ y_{1}O_{D}

is a good lattice in V_{1} for each 0 ≤ i ≤ n. Moreover, B_{i} = A_{0} + N_{i} is a good lattice in V.

1.6.2. Regular self-dual lattice chains and barycenters. Two lattices L_{1}, L_{2} in V are said to be similar if L_{1} = L_{2}ϖ_{D}^{k} for some integer k. A non-empty collection of lattices L = {L_{i}}_{i∈Z} in V is called a lattice chain if it satisfies the following conditions:

(i) L is totally ordered by inclusion i.e., L_{i} ⊆ L_{i+1} for each i; (ii) each lattice L_{i} is similar to a good lattice or the dual lattice of a good lattice; (iii) L ∈ L implies that the whole similarity class of L is contained in L i.e., if L ∈ L then

Lϖ^{k} ∈ L for any k; (iv) there exists a number m such that L_{i+m} = L_{i}ϖ for all i. Let L be a lattice chain, then the number m is called the period of L. L is said to be regular if L_{i} ≠ L_{j} whenever i ≠ j. Let m be a positive integer and m_{0} = 0 or 1. A lattice chain L = {L_{i}}_{i∈Z} is said to be self-dual with numerical invariant (m, m_{0}) if it has period m and L_{i}^{∗} = L_{−i−m_{0}} for all i. We have a bijection between the following two sets:

(1) regular self-dual lattice chains in V;

(2) (genuine) barycenters in a chamber. For each parahoric subgroup we have a standard filtration defined by Prasad-Raghunathan. This filtration is equal to the Moy-Prasad filtration at the corresponding barycenter.

1.6.3. Small admissible lattice chains. A lattice in V is called a big (resp. small) critical lattice if it is similar to B_{0}^{∗} or B_{n} (resp. B_{0} or B_{n}^{∗}) defined in subsection 1.6.1. L is called a small (resp. big) admissible lattice chain with numerical invariant (m, m_{0}) if it has period m and satisfies one of the following conditions:

(i) when m is even and m_{0} = 1, L_{i}^{∗} = L_{−i−1} for all i; (ii) when m is even and m_{0} = 0, L_{i}^{∗} = L_{−i} for all i ≢ 0 or m/2 (mod m) and L_{i} is small

(res. big) critical if i ≡ 0 or m/2 (mod m); (iii) when m is odd and n_{0} = 1, L_{i}^{∗} = L_{−i−1} for all i ≢ 0 (mod m) and L_{i} is small (res. big)

critical if i ≡ 0 (mod m); (iv) when m is odd and m_{0} = 0, L_{i}^{∗} = L_{−i} for all i ≢ (m − 1)/2 (mod m) and L_{i} is small

(res. big) critical if i ≡ (m − 1)/2 (mod m).

1.7. Filtration subgroups by lattice chains. Let L be a regular small admissible lattice chain in V with numerical invariant (m, m_{0}). Define L^{♯} := { L_{i}^{♯} | L_{i}^{♯} := L_{−i−m_{0}}^{∗}}_{i∈Z}. Then we define

\{ \begin{matrix}G_{L,d/m} := \{ g ∈ G | (g − 1).L_{i} ⊆ L_{i+d}, (g − 1).L_{i}^{♯} ⊆ L_{i+d}^{♯} for all i \} & for d ≥ 1; \\ G_{L,(d/m)^{+}} := \{ g ∈ G | (g − 1).L_{i} ⊆ L_{i+d+1}^{♯}, (g − 1).L_{i}^{♯} ⊆ L_{i+d} for all i \} & for d ≥ 0. \end{matrix}

If L is self-dual then L_{i} = L_{i}^{♯}, and G_{L,d/m} = { g ∈ G | (g − 1).L_{i} ⊆ L_{i+d} for all i } and G_{L,(d/m)^{+}} = G_{L,(d+1)/m}.

1.7.1. Similarly, we can define the filtration 𝔤_{L,d/m}, 𝔤_{L,(d/m)}, 𝔤_{L,−d/m}^{∗}, and 𝔤_{L,(−d/m)^{+}}^{∗}.

1.7.2. Comparison of two filtrations. Let G = U(V) be a classical group. Then essentially we have a bijective correspondence between the following two sets:

(1) regular small admissible lattice chains in V;

(2) generalized barycenters in a chamber.

Proposition. Let x be a generalized barycenter, L be the corresponding regular small admissible lattice chain and m be the period of L, {r_{i}}_{i≥0} be the sequence associated x in subsection 1.1. Then

(i) If L is self-dual, then r_{i} = i/m for any i. If L is not self-dual, then r_{i} = i/2m. (ii) Every residue class in 𝔤_{x,−r_{i}}^{∗}/𝔤_{x,−r_{i−1}}^{∗} contains a nilpotnet element if L is not self-dual,

m ≥ 2, and i is odd. (iii) G_{x,i/m} = G_{L,i/m} and G_{x,(i/m)^{+}} = G_{L,(i/m)^{+}}.

1.8. We have the following refinement of Theorem 1.5.1 and Proposition 1.7.2.

Theorem. Suppose that the local Dynkin diagram of G does not have any ramified ends, then an irreducible admissible representation (π, V) of positive depth of G contains a minimal K-type of the form (G_{L,d/m}, χ), where L is a regular small admissible lattice chain of period m such that L, d, m satisfy one of the following two conditions:

(i) gcd(d, m) = 1; or (ii) gcd(d, m) = 2 and L is self-dual.

2. Local theta correspondence 2.1. Lattice model of the Weil representations. From now on, we assume that the characteristic of f is odd. Let W be a finite dimensional symplectic space over F, with the skew-symmetric bilinear form ⟨⟨, ⟩⟩. Let ψ be a fixed non-trivial (addivtive) character of F of conductor O. Let A_{0} be a self-dual lattice in W. Let S(A_{0}) be the space of locally constant, compactly supported functions f: W → C satisfying the condition

f(a + w) = ψ(\frac{1}{2}⟨⟨w, a⟩⟩)f(w),

where a ∈ A_{0}, and w ∈ W. The (projective) action ω_{ψ} of Sp(W) on the space S(A_{0}) is given by

(ω_{ψ}(g) · f)(w) = ∫_{A_{0}} ψ(\frac{1}{2}⟨⟨w, a⟩⟩)f(g^{−1}.(w + a)) da,

where g ∈ Sp(W), f ∈ S(A_{0}), w ∈ W, and da is some Haar measure on A_{0}. We can lift ω_{ψ} to get a representation, also denoted by ω_{ψ}, of the metaplectic cover S̃p(W) of Sp(W). (ω_{ψ}, S(A_{0})) is called a lattice model of the Weil representation of S̃p(W). If B is a union of A_{0}-cosets, let S(A_{0})_{B} denote the subspace of functions supported in B. The (one-dimensional) space S(A_{0})_{A_{0}+w} is also denoted by S(A_{0})_{w}.

2.2. Reductive dual pairs. Let (V_{1}, ⟨, ⟩_{1}) be a right ϵ_{1}-hermitian D-space, (V_{2}, ⟨, ⟩_{2}) a right ϵ_{2}-hermitian D-space where ϵ_{1}, ϵ_{2} = 1, −1 and ϵ_{1}ϵ_{2} = −1. Let W := V_{1} ⊗_{D} V_{2} and ⟨⟨, ⟩⟩ := Trd_{D/F} (⟨, ⟩_{1} ⊗ τ ∘ ⟨, ⟩_{2}) where Trd_{D/F} denotes the reduced trace from D to F. Then (U(V_{1}), U(V_{2})) forms a reductive dual pair in Sp(W). The reductive dual pair (U(V_{1}), U(V_{2})) is called unramified if D is F or a unramified quadratic extension of F and there exist self-dual lattices in both spaces (V_{1}, ⟨, ⟩_{1}), (V_{2}, ⟨, ⟩_{2}).

2.3. Local theta correspondence. Let Ũ(V_{i}) be the inverse image of U(V_{i}) in S̃p(W) under the map S̃p(W) → Sp(W). By restricting the Weil representation ω_{ψ} to Ũ(V_{1}) · Ũ(V_{2}), there is a correspondence between some irreducible admissible representations of Ũ(V_{1}) and those of Ũ(V_{2}). It is proved by Howe and Waldspurger that this correspondence is one-to-one and is called the local theta correspondence.

2.3.1. We know that the map K̃ → K splits for any compact subgroup K of U(V_{i}). By fixing an embedding of the Iwahori subgroup I_{i} → Ũ(V_{i}) the definitions of minimal K-types and depths can be generalized to any irreducible admissible representation of the group Ũ(V_{i}).

2.4. Let (U(V_{1}), U(V_{2})) be a reductive dual pair. Let Γ_{M} (resp. Γ′_{M}) be a fixed maximal good lattice in V_{1} (resp. V_{2}). Let (M, N) be a pair of lattices in V_{1} such that

Nϖ_{D} ⊆ M ⊆ N
∩ ∩ ∩
Γ_{M}^{∗}ϖ_{D} ⊆ Γ_{M} ⊆ Γ_{M}^{∗}.

Define J_{M,N} := { g ∈ U(V_{1}) | (g − 1).N^{∗} ⊆ M }, and B_{M,N} := M^{∗} ⊗_{O_{D}} Γ′_{M} + N^{∗} ⊗_{O_{D}} Γ_{M}^{′∗}.

Theorem (Waldspurger). Let A_{0} be a self-dual lattice in W contained in B_{M,N}, then

S(A_{0})^{J_{M,N}} = ω_{ψ}(H_{2}).(S(A_{0})_{B_{M,N}})^{J_{M,N}} ,

where H_{2} is the Hecke algebra of Ũ(V_{2}).

2.5. Suppose that we have small admissible lattice chains L = {L_{i}}_{i∈Z}, L′ = {L′_{i}}_{i∈Z} in V_{1}, V_{2} with numerical invariants (m, m_{0}), (m′, m′_{0}) respectively. Let d be a non-negative integer. We assume that m = m′ and m + m_{0} + m′_{0} + d is even. Then define

B(L, L′, d/m) = ∑_{i+j=(m−2−m_{0}−m′_{0}−d)/2} L_{i} ⊗_{O_{D}} L′_{j}.

B(L, L′, d/m) is a lattice in W containing a self-dual lattice.

2.6. Theorem. Assume that d ≥ 0, gcd(d, m) = 1, or gcd(d, m) = 2 and m_{0} = 1. Put m′_{0} = 0 or 1 so that m + m_{0} + m′_{0} + d is even. Let A_{0} be a self-dual lattice contained in B(L, L′, d/m). Then

S(A_{0})^{G_{L,(d/m)^{+}}} = ω_{ψ}(H_{2}).(∑_{L′} S(A_{0})_{B(L,L′,d/m)}),

where the sum is taken over all small admissible lattice chains L′ with numerical invariant (m, m′_{0}).

3. Moment maps and minimal K-types 3.1. Trace forms. Let (U(V_{1}), U(V_{2})) be a reductive dual pair. Let 𝔅_{1} (resp. 𝔅_{2}) denote the trace form of the Lie algebra 𝔤 (resp. 𝔤′) of U(V_{1}) (resp. U(V_{2})) i.e., 𝔅_{i}(X, Y) = Trd_{D/F} (tr(XY)). For x, y ∈ V_{i}, define c_{x,y}: V_{i} → V_{i} by c_{x,y}(v) = x⟨y, v⟩_{i} − ϵ_{i}y⟨x, v⟩_{i}. Then c_{x,y} ∈ 𝔤 (or 𝔤′). It is not difficult to check that

𝔅_{i}(c_{x,y}, X) = Trd_{D/F} (⟨−X.y, x⟩_{i} + ϵ_{i}⟨X.x, y⟩_{i}),

where X in 𝔤 (resp. 𝔤′).

3.1.1. Moment maps. As usual let W = V_{1} ⊗_{D} V_{2} and 𝔤^{∗} (resp. 𝔤^{′∗}) be the dual space of the Lie algrbra 𝔤 (resp. 𝔤′). We have the moment maps 𝔐: W → 𝔤^{∗} and 𝔐′: W → 𝔤^{′∗}. Now we identify 𝔤 (resp. 𝔤′) with 𝔤^{∗} (resp. 𝔤^{′∗}) by the trace form 𝔅_{1} (resp. 𝔅_{2}). Therefore we may regard c_{x,y} as an element in 𝔤^{∗} (resp. 𝔤^{′∗}) for x, y ∈ V_{1} (resp. V_{2}). If w = ∑_{i} a_{i} ⊗ b_{i} ∈ W, where a_{i} ∈ V_{1}, b_{i} ∈ V_{2}, then 𝔐(w) = ∑_{i,j} c_{a_{i},a_{j}⟨b_{j},b_{i}⟩_{2}} and 𝔐′(w) = ∑_{i,j} c_{b_{i},b_{j}⟨a_{j},a_{i}⟩_{1}}. We can prove that if w ∈ B(L, L′, d/m) then 𝔐(w) ∈ 𝔤_{L,−d/m}^{∗}. Let 𝔐̄(w) = 𝔐(w) + 𝔤_{L,(−d/m)^{+}}^{∗} ∈ 𝔤_{L,−d/m}^{∗}/𝔤_{L,(−d/m)^{+}}^{∗}.

3.2. Let L = {L_{i}}_{i∈Z} be a regular small admissible lattice chain in V_{1} and (π_{1}, V_{1}) be an irreducible admissible representation of Ũ(V_{1}) with non-trivial G_{L,(d/m)^{+}}-fixed vectors for some d > 0. Suppose that there is a vector w ∈ B(L, L′, d/m) such that S(A_{0})_{w} has non-trivial image under the map Π: S(A_{0}) → V_{1} ⊗_{C} V_{2}. Now we have an action of G_{L,d/m} on the one-dimensional space Π(S(A_{0})_{w}). Then we have a character denoted by ψ_{w} of G_{L,d/m}/G_{L,(d/m)^{+}} on Π(S(A_{0})_{w}). Because (G_{L,d/m}/G_{L,(d/m)^{+}})^{∧} ≃ 𝔤_{L,−d/m}^{∗}/𝔤_{L,(−d/m)^{+}}^{∗}, there exists an element X ∈ 𝔤_{L,−d/m}^{∗} such that the coset X + 𝔤_{L,(−d/m)^{+}}^{∗} corresponds to the character ψ_{w} of the quotient group G_{L,d/m}/G_{L,(d/m)^{+}} on Π(S(A_{0})_{w}).

3.2.1. Now this character ψ_{w} is given by g ↦ ψ(\frac{1}{2}⟨⟨(g − 1).w, w⟩⟩), where g ∈ G_{L,d/m}. Assume that w = ∑_{i} γ_{i} ⊗ η_{i}, where γ_{i} ∈ V_{1}, η_{i} ∈ V_{2}, then we have

ψ(\frac{1}{2}⟨⟨(g − 1).(∑_{i} γ_{i} ⊗ η_{i}), ∑_{i} γ_{i} ⊗ η_{i}⟩⟩) = ψ(𝔅_{1}(∑_{i,j} −\frac{1}{4}c_{γ_{i},γ_{j}⟨η_{j},η_{i}⟩_{2}}, g − 1)).

Therefore the element X in subsection 3.2 can be chosen to be −\frac{1}{4} ∑_{i,j} c_{γ_{i},γ_{j}⟨η_{j},η_{i}⟩_{2}} i.e., −\frac{1}{4}𝔐(w).

3.3. For a given small admissible lattice chain L of period m in V, let v_{i} = L_{i}/L_{i+1}^{♯}. Identify L_{i}/L_{i+1}^{♯} with L_{i+m}/L_{i+1+m}^{♯} via multiplication by ϖ_{D}. Let v = ⨁_{i=1}^{m} v_{i}, then v is a space over the residue field of D and graded by Z/mZ. Define v′_{i′} = L′_{i′}/L_{i′+1}^{′♯}, v′ = ⨁_{i=1}^{m} v′_{i} similarly. Now we know that 𝔐̄(w) is a graded homomorphism of v of degree −d i.e., 𝔐̄(w)(v_{k}) ⊆ v_{k−d} for all k.

Proposition. 𝔐̄(w) is a nilpotent endomorphism of v if and only if 𝔐̄′(w) is a nilpotent endomorphism of v′.

3.4. If L′ is a small admissible lattice chain but not a regular lattice chain, then some v′_{i} is trivial. If we also assume that gcd(d, m) = 1, then it is clear that 𝔐̄′(w) is a nilpotent endomorphism of v′. Therefore we have the following proposition.

Proposition. Suppose that w ∈ B(L, L′, d/m) for some regular small admissible lattice chain L and a small admissible lattice chain L′. Assume that gcd(d, m) = 1, and the residue class −\frac{1}{4}𝔐(w) + 𝔤_{L,(−d/m)^{+}}^{∗} contains no nilpotent element, then any two lattices in L′ are distinct i.e., L′ is a regular small admissible lattice chain.

Therefore we need only to take care of the case gcd(d, m) = 2 by Theorem 1.8.

3.4.1. In this subsection we assume that gcd(d, m) = 2. The idea is: if L′ is not a lattice chain, then we can find regular lattice chains M, M′ of period m/2 such that B(L, L′, d/m) ⊆ B(M, M′, \frac{d}{2}/\frac{m}{2}). Therefore we can reduce the problem to situation in Subsection 3.4.

3.5. Theorem. Let (G, G′) = (U(V_{1}), U(V_{2})) be a reductive dual pair. Let (π_{1}, V_{1}) (resp. (π_{2}, V_{2})) be an irreducible admissible representation of Ũ(V_{1}) (resp. Ũ(V_{2})) such that V_{1} and V_{2} are paired by the local theta correspondence. Then the depth of V_{1} is equal to the depth of V_{2}.

Now the proof of Theorem 3.5 becomes clear. At least one of the local Dynkin diagrams of U(V_{1}), U(V_{2}) does not have ramified ends. We assume the local Dynkin diagram of U(V_{1}) satisfies this condition, so we can apply Theorem 1.5.1 and Proposition 1.7.2 to describe the minimal K-types of irreducible admissible representations of Ũ(V_{1}). If we assume that V_{1}^{G_{L,(d/m)^{+}}} ≠ {0}, then there is a regular small admissible lattice chain L′ of period m or m/2 (the latter happens only when m is even) in V_{2} such that V_{2}^{G_{L′,(d/m)^{+}}} (or

V_{2}^{G_{L′,( \frac{d}{2} / \frac{m}{2} )^{+}}} ) is not trivial. Finally results in subsection 3.2.1 and 3.3 enforce the depth of π_{1} has to be equal to the depth of π_{2}.

3.6. Therefore we have the following theorem immediately from Theorem 3.5 and the stable range of the local theta correspondence. This theorem complete our analysis on minimal K-types for all p-adic classical groups.

Theorem. Let (π, V) be an irreducible admissible representation of a p-adic classical group G, then (π, V) has a minimal K-type (G_{x,r}, χ) where x is a generalized barycenter in the Bruhat-Tits building of G.

3.7. The depths for classical groups. Let ℓ_{G} be the maximal period of all small admissible lattice chains in V, then the depth of an irreducible admissible representation of a classical group G will be of the form s/t, where t ≤ ℓ_{G}. The number ℓ_{G} can be computed easily for any G, for example ℓ_{G} = 2n for G = Sp_{2n}(F).

Corollary. If (π, V) is an irreducible admissible representation of a classical group G of depth s/ℓ_{G} such that gcd(s, ℓ_{G}) = 1, then π is supercuspidal.

Of course, this corollary is well known when G = GL_{n}(F), in which case ℓ_{G} is just equal to n.

3.8. Cases for depth zero. The following corollary connects a simple relation between the local theta correspondence over a p-adic filed and the local theta correspondence over its residue field. Here we will write G_{L,r} instead of G_{L,r} if L is a regular self-dual lattice chain of period 1 or 2 determined by a single lattice L. It is not difficult to see: if a representation (π, V) of U(V) is of depth zero, then V^{G_{L,0^{+}}} ≠ {0} for some good lattice L in V.

3.8.1. Let L (resp. L′) be a good lattice in V_{1} (resp. V_{2}). Let B = B(L, L′) = L^{∗} ⊗ L′ + L ⊗ L^{′∗}. It is easy to check that Bϖ_{F} ⊆ B^{∗} ⊆ B. Then B/B^{∗} can be identified as a vector space over f with a non-degenerated skew-symmetric bilinear form. Hence we can consider the finite symplectic group Sp(B/B^{∗}) and its Weil representation. Let K_{B} be the stabilizer of B in Sp(W) and K′_{B} := { g ∈ K_{B} | (g − 1).B ⊆ B^{∗} }. Now we have a close relation between the Weil representation of (the metaplectic cover of) a p-adic symplectic group and the Weil representation of a finite symplectic group as follows.

Proposition. Let S be the Weil representation of S̃p(W), then the action of K_{B}/K′_{B} ≃ Sp(B/B^{∗}) on S^{K′_{B}} is the Weil representation.

3.8.2. Proposition. The pair (G_{L}/G_{L,0^{+}}, G′_{L′}/G′_{L′,0^{+}}) is a reductive dual pair in the finite symplectic group Sp(B/B^{∗}) defined over the residue field f of F.

3.8.3. Corollary. Let (G, G′) = (U(V_{1}), U(V_{2})) be a reductive dual pair. Let (π_{1}, V_{1}) be an irreducible smooth representation of Ũ(V_{1}) and (π_{2}, V_{2}) be an irreducible smooth representation of Ũ(V_{2}) such that two representations correspond in the local theta correspondence. Then V_{1}^{G_{L,0^{+}}} ≠ {0} for some good lattice L in V_{1} if and only if there is

a good lattice L′ in V_{2} such that V_{2}^{G′_{L′,0^{+}}} ≠ {0}. In particular, depth(π_{1}) = 0 if and only if depth(π_{2}) = 0. Moreover if ϕ_{1} is an irreducible representation of G_{L}/G_{L,0^{+}} which occurs in V_{1}^{G_{L,0^{+}}} , then there is an irreducible representation ϕ_{2} of G′_{L′}/G′_{L′,0^{+}} which oc-

curs in V_{2}^{G′_{L′,0^{+}}} such that ϕ_{1} and ϕ_{2} correspond in the correspondence for the dual pair (G_{L}/G_{L,0^{+}}, G′_{L′}/G′_{L′,0^{+}}).

3.8.4. Corollary. Let (U(V_{1}), U(V_{2})) be a reductive dual pair. Let I_{i} be the Iwahori subgroup of U(V_{i}). And Let (π_{i}, V_{i}) be an irreducible admissible representation of Ũ(V_{i}), i = 1, 2. Suppose that π_{1}, π_{2} are paired by local theta correspondence, then V_{1}^{I_{1}} ≠ {0} if and only if V_{2}^{I_{2}} ≠ {0}.

Corollary 3.8.4 has been proved by A.-M. Aubert for unramified reductive dual pairs.

4. References

[MVW] C. Mœglin, M.-F. Vigneras and J.-L. Waldspurger, Correspondances de Howe sur un corps p-adigues, Lect. Note. Math. 1291, Springer-Verlag, Berlin-Heidelberg-New York, 1987.

[MP1] A. Moy and G. Prasad, Unrefined minimal K-types for p-adic groups, Invent. Math. 116 (1994), 393–408.

[MP2] A. Moy and G. Prasad, Jacquet functors and unrefined minimal K-types, Com-

ment. Math. Helv. 71 (1996), 98–121.

[P] S.-Y. Pan, local theta correspondence and unrefined minimal K-types, preprint, 1997.

[PY] S.-Y. Pan and J.-K. Yu, Unrefined minimal K-types for p-adic classical groups, preprint, 1997.

[W] J.-L. Waldspurger, Demonstration d’une conjecture de duality de Howe dans le case p-adiques, p ≠ 2, in Festschrift in honor of I. Piatetski-Shapiro, Israel Math. Conf. Vol. 2 (1990), 267–324.

Shu-Yen Pan Department of Mathematics Cornell University Ithaca, NY 14853

E-mail: pan@math.cornell.edu