Wee Teck Gan

Representations of metaplectic groups I: epsilon dichotomy and local Langlands correspondence

Using theta correspondence, we classify the irreducible representations of Mp 2 n in terms of the irreducible representations of SO 2 n +1 and determine many properties of this classification. This is a local Shimura correspondence which extends the well-known results of Waldspurger for n =1.

Group schemes and local densities

The subject matter of this paper is an old one with a rich history, beginning with the work of Gauss and Eisenstein, maturing at the hands of Smith and Minkowski, and culminating in the fundamental results of Siegel. More precisely, if L is a lattice over Z (for simplicity), equipped with an integral quadratic form Q, the celebrated Smith-Minkowski-Siegel mass formula expresses the total mass of (L,Q), which is a weighted class number of the genus of (L,Q), as a product of local factors. These local factors are known as the local densities of (L,Q). Subsequent work of Kneser, Tamagawa and Weil resulted in an elegant formulation of the subject in terms of Tamagawa measures. In particular, the local density at a non-archimedean place p can be expressed as the integral of a certain volume form ωld over AutZp(L,Q), which is an open compact subgroup of AutQp(L,Q). The question that remains is whether one can find an explicit formula for the local density. Through the work of Pall (for p 6= 2) and Watson (for p = 2), such an explicit formula for the local density is in fact known for an arbitrary lattice over Zp (see [P] and [Wa]). The formula is obviously structured, though [CS] seems to be the first to comment on this. Unfortunately, the known proof (as given in [P] and [K]) does not explain this structure and involves complicated recursions. On the other hand, Conway and Sloane [CS, §13] have given a heuristic explanation of the formula. In this paper, we will give a simple and conceptual proof of the local density formula, for p 6= 2. The view point taken here is similar to that of our earlier work [GHY], and the proof is based on the observation that there exists a smooth affine group scheme G over Zp with generic fiber AutQp(L,Q), which satisfies G(Zp) = AutZp(L,Q). This follows from general results of smoothening [BLR], as we explain in Section 3. For the purpose of obtaining an explicit formula, it is necessary to have an explicit construction of G. The main contribution of this paper is to give such an explicit construction of G (in Section 5), and to determine its special fiber (in Section 6). Finally, by comparing ωld and the canonical volume form ωcan of G, we obtain the explicit formula for the local density in Section 7. The smooth group schemes constructed in this paper should also be of independent interest.

On an exact mass formula of Shimura

In a series of recent papers, G. Shimura obtained an exact formula for the mass of a maximal lattice in a quadratic or hermitian space over a totally real number field. Using Bruhat-Tits theory, we obtain a quick and more conceptual proof of his formula when the form is totally definite.

On endoscopy and the refined Gross–Prasad conjecture for (SO 5 , SO 4 )

We prove an explicit formula for periods of certain automorphic forms on SO5 × SO4 along the diagonal subgroup SO4 in terms of L-values. Our formula also involves a quantity from the theory of endoscopy, as predicted by the refined Gross-Prasad conjecture.

Equidistribution of Integer Points on a Family of Homogeneous Varieties: A Problem of Linnik

We study an equidistribution problem of Linnik using Hecke operators.

The Gross–Prasad conjecture and local theta correspondence

We establish the Fourier–Jacobi case of the local Gross–Prasad conjecture for unitary groups, by using local theta correspondence to relate the Fourier–Jacobi case with the Bessel case established by Beuzart-Plessis. To achieve this, we prove two conjectures of Prasad on the precise description of the local theta correspondence for (almost) equal rank unitary dual pairs in terms of the local Langlands correspondence. The proof uses Arthur’s multiplicity formula and thus is one of the first examples of a concrete application of this “global reciprocity law”.

Haar Measure and the Artin Conductor

Let G be a connected reductive group, defined over a local, non-archimedean field k. The group G(k) is locally compact and unimodular. In [Gr], a Haar measure |ωG| was defined on G(k), using the theory of Bruhat and Tits. In this note, we give another construction of the measure |ωG|, using the Artin conductor of the motive M of G over k. The equivalence of the two constructions is deduced from a result of Prasad [P]. 1. The Root Datum and Motive of G In this section, k is an arbitrary field and G is a connected reductive group over k. We let k be an algebraic closure of k, ks the separable closure of k in k, and Γ = Gal(ks/k). Let T ⊂ B ⊂ G be a maximal torus, contained in a Borel subgroup, defined over ks. Let Ψ = Ψ(G,B, T ) be the based root datum defined by this choice. We recall (cf [Sp], pg3-12) that: Ψ = (X(T ),∆(T,B),X•(T ),∆•(T,B)) , (1.1) with X•(T ) and X•(T ) the character and cocharacter groups of T respectively, and ∆• and ∆• the simple roots and coroots determined by B respectively. Let W = NG(T )/T be the Weyl group of Ψ. The finite group W acts as automorphisms of X•(T ), and is generated by the reflections: sα(x) = x− 〈x, α〉α (1.2) for α ∈ ∆•. The Galois group Γ acts as automorphisms of Ψ, ie as automorphisms of the group X•(T ) preserving the finite set ∆•, as follows. If σ ∈ Γ, then we can find g ∈ G(ks) such that: Int(g)(σT ) = gσ(T )g = T, and Int(g)(σB) = gσ(B)g = B, with g well-defined up to left multiplication by T (ks). Hence it induces a well-defined automorphism ψ(σ) : X(T ) −→ X(T ) Date: 14th January 1997. 1 2 BENEDICT H. GROSS AND WEE TECK GAN preserving ∆•. Hence we get a group homomorphism ψ : Γ −→ Aut(Ψ). Via ψ, Γ acts on Aut(Ψ) by inner automorphisms. Similarly, if f : G −→ G is any automorphism of G over ks, it induces an automorphism ψ(f) of Ψ, which depends only on the image of f in the quotient group Outks(G) of outer automorphisms. The resulting map Outks(G) −→ Aut(Ψ) is an isomorphism which respects the respective Galois actions on the two groups (cf [Sp], pg10). The Galois group Γ also acts on W , via the formula: σ(sα) = sσ(α) (1.3) and the semi-direct product W ⋊ Γ acts on the rational vector space E = X(T )⊗Q (1.4) Let R = Sym•(E)W , which is a graded Q[Γ]-module. Let R+ be the ideal of elements of positive degree in R, and define: V = R+/R 2 + = ⊕d≥1Vd (1.5) This is a graded Q[Γ]-module, and Chevalley proved that dim(V ) = dim(E) (cf [Ch]). Steinberg extended the proof to show that E and V are isomorphic Γ-modules (cf [St], pg22). We sketch a proof of this result that does not involve the classification of irreducible root systems. Proposition 1.6. The Q[Γ]-modules E and V are isomorphic. Proof. By the criterion in [Se, pg 104], it suffices to show that for all σ ∈ Γ, the fixed spaces Eσ and V σ have the same dimension. For any graded Γ-module A = ⊕Am, we define the Poincare series of σ by: P (A,σ)(t) = ∑ tr(σ|Am)t. Then P (A ⊗ B) = P (A)P (B). Steinberg showed that there is an isomorphism of graded Γ-modules: S(E) ∼= S(⊕Vd)⊗A Here A is finite dimensional, with basis {bw}w∈W , and Γ-action given by: σ(bw) = bσ(w) The degree of bw is the length l(w) of w, with respect to the generators sα furnished by ∆•. This isomorphism yields the following identity of Poincare series: det(1− σt|E) = ∏ d≥1 det(1 − σt|Vd) · ∑

The Saito-Kurokawa Space of PGSp4 and Its Transfer to Inner Forms

We discuss the construction, characterization and classification of the Saito-Kurokawa representations of PGS{inp4} and its inner forms, interpreting them in the framework of Arthur’s conjectures. These Saito-Kurokawa representations are among the first examples of the so-called CAP representations or shadows of Eisenstein series.

On the regularized Siegel–Weil formula (the second term identity) and non-vanishing of theta lifts from orthogonal groups

Abstract We derive a (weak) second term identity for the regularized Siegel–Weil formula for the even orthogonal group, which is used to obtain a Rallis inner product formula in the “second term range”. As an application, we show the following non-vanishing result of global theta lifts from orthogonal groups. Let π be a cuspidal automorphic representation of an orthogonal group O(V) with dimV = m even and r + 1 ≦ m ≦ 2r. Assume further that there is a place ν such that πν ≅ πν ⊗ det. Then the global theta lift of π to Sp2r does not vanish up to twisting by automorphic determinant characters if the (incomplete) standard L-function LS (s, π) does not vanish at s = 1 + (2r – m)/2. Note that we impose no further condition on V or π. We also show analogous non-vanishing results when m > 2r (the “first term range”) in terms of poles of LS (s, π) and consider the “lowest occurrence” conjecture of the theta lift from the orthogonal group.

ON SHALIKA PERIODS AND A THEOREM OF JACQUET-MARTIN

Let