GERMS OF CHARACTERS

Fiona Murnaghan

University of Toronto

Let F be a p-adic field of characteristic zero, G a a reductive group defined over F, and G = G(F). If π is an irreducible admissible representation of G, Θ_{π} denotes the character of π, viewed either as a distribution or as a locally integrable function which is locally constant on the set G_{reg} of regular semisimple elements in G. By the germ of Θ_{π} (at the identity), we mean the restriction of (the function) Θ_{π} to V ∩ G_{reg}, where V is some open neighbourhood of the identity in G. In this talk, we discuss the relation between germs of characters and linear combinations of Fourier transforms of measures on Ad G-orbits in the Lie algebra 𝔤 of G.

The symbol O will be used for an Ad G-orbit in 𝔤. Given X ∈ 𝔤, O_{G}(X) or O(X) will be used for the Ad G-orbit of X in 𝔤. Let C_{c}^{∞}(𝔤) be the set of of locally constant compactly supported complex valued functions on 𝔤. Given an orbit O and a function f ∈ C_{c}^{∞}(𝔤), let μ_{O}(f) denote the integral of f with respect to a G-invariant measure on O.

Recall that if B(·, ·) is a non-degenerate symmetric G-invariant bilinear form on 𝔤, ψ is a non trivial character of F, and dX is a Haar measure on 𝔤, the Fourier transform f̂ of f is given by:

f̂(X) = ∫𝔤 ψ((B(X, Y)) f(Y) dY, f ∈ C_{c}^{∞}(𝔤).

Given an Ad G-orbit O, the Fourier transform μ̂_{O} of the distribution μ_{O} is defined by μ̂_{O}(f) = μ_{O}(f̂), f ∈ C_{c}^{∞}(𝔤). The distribution μ̂_{O} is given by a locally integrable function on 𝔤 which is locally constant on 𝔤_{reg} (regular semisimple elements in 𝔤). This function will also be denoted by μ̂_{O}.

Let π be an irreducible admissible representation of G. Recall ([HC]) that Harish-Chandra’s local character expansion (at the identity) expresses the germ of Θ_{π} as a linear combination of { μ̂_{O} | O ∈ (N_{G}) }, where (N_{G}) is the set of nilpotent Ad G-orbits: there exists an open neighbourhood V_{π} of zero in 𝔤, and complex numbers c_{O}(π), O ∈ (N_{G}), such that

Θ_{π}(exp X) = ∑_{O∈(N_{G})} c_{O}(π) μ̂_{O}(X), X ∈ V_{π} ∩ 𝔤_{reg}.

Here exp denotes the exponential map. In some examples a different map can be used in place of exp. For example, if G = GL_{n}(F), resp. G is a classical group, X ↦ 1 + X, resp. the Cayley transform, is sometimes used.

It can be difficult to compute the coefficients c_{O}(π) and the functions μ̂_{O}. For certain representations π, it might be convenient to express the germ of Θ_{π} in terms of Fourier transforms of non nilpotent Ad G-orbits. Suppose that π is a tamely ramified irreducible supercuspidal representation of a classical group ([A], [H], [Mo 1-3], [My1]). As shown in [Mu1-4], for many such π, there exists a regular elliptic s_{π} ∈ 𝔤 such that

Θ_{π}(exp X) = d(π) μ̂_{O(s_{π})}(X), X ∈ V ∩ 𝔤_{reg},

for some open neighbourhood V of zero in 𝔤. Here, d(π) is the formal degree of π. For convenience, we could renormalize the measure on O(s_{π}) so that d(π) = 1. The orbit O(s_{π}) is defined using the construction of π via induction from a representation of some open compact mod centre subgroup of G. That is, O(s_{π}) depends on the K-type(s) of π. The relation of the above to the local character expansion is given by:

c_{O}(π) = d(π) Γ_{O}(s_{π}), O ∈ (N_{G}).

Here, Γ_{O} is the Shalika germ corresponding to the nilpotent orbit O.

Let π be an irreducible admissible representation of G. We are interested in expressing the germ of Θ_{π} as a linear combination of { μ_{O} | O ∈ S_{π} }, where S_{π} is some finite collection of Ad G-orbits in 𝔤. Of course there may be several choices for S_{π}. It follows from Harish-Chandra’s local character expansion that we can always take S_{π} = (N_{G}). For the supercuspidal representations mentioned above, we can also take S_{π} = { O(s_{π}) }. Given a semisimple element s ∈ 𝔤, let Ω_{G}(s) denote the set of Ad G-orbits O ⊂ 𝔤 which contain elements whose semisimple part is conjugate to s; that is, orbits O whose closure contains s. In particular, Ω_{G}(0) = (N_{G}). We will discuss some cases where S_{π} = Ω_{G}(s) for some non zero s. A different approach would be to look for an S_{π} consisting of orbits in 𝔤_{reg}.

Results which we use include:

1. Hecke algebra isomorphisms and K-types

2. Uniformity of local character expansions - relation between the depth of π and the size of the neighbourhood V_{π}.

3. Descent properties of orbital integrals. The idea is to use 1–3 to relate the germ of Θ_{π} to the germ of Θ_{π′} for an irreducible admissible representation π′ of a reductive subgroup G′ of G. In particular, if we know the local character expansion for π′, what does that imply about the germ of Θ_{π}? More generally, if we have some collection S_{π′} as above for π′, do we get a corresponding collection S_{π} for π? If so, what is the relation between the coefficients in the two linear combinations? This is work in progress. We will discuss examples.

Example 1:

• G = GL_{n}(F)

• d a positive integral divisor of n

• E/F an unramified extension of degree d

• ε_{d} a root of unity in O_{E} whose image in the residue class field 𝔼 = O_{E}/𝔭_{E} generates 𝔼/𝔽, where 𝔽 = O_{F} /𝔭_{F}

• M_{d} = GL_{d} × ⋯ × GL_{d} (n/d times), viewed as a standard Levi subgroup of GL_{n}, and M_{d} = M_{d}(F)

• s_{d} = (ε_{d}, …, ε_{d}) ∈ M_{d} ⊂ G, ε_{d} identified with an element of GL_{d}(F) via a basis for E/F consisting of roots of unity

• G′ ≃ GL_{n/d}(E), the centralizer of s_{d} in G

• K_{d} the parahoric subgroup of G which is equal to the inverse image in GL_{n}(O_{F}) of a standard parabolic subgroup of GL_{n}(𝔽) having Levi component M_{d}(𝔽)

• K_{d}^{u} the pro unipotent radical of K_{d}

• σ an irreducible cuspidal representation of GL_{d}(𝔽)

• ρ the pullback to K_{d} of the representation (⊗σ)^{n/d} of M_{d}(𝔽) ≃ K_{d}/K_{d}^{u}

• π an admissible representation of G such that the restriction of π to K_{d} contains ρ, written (K_{d}, ρ) ↪ π.

• H(G//K_{d}, ρ) the ρ-spherical Hecke algebra of compactly supported functions f from G to the endomorphisms of the space of ρ such that

f(k_{1}xk_{2}) = ρ(k_{1})f(x)ρ(k_{2}), ∀ k_{1}, k_{2} ∈ K_{d}, x ∈ G.

• I′ an Iwahori subgroup of G′

• H(G′//I′) = H(G′/I′, 1_{I′}), 1_{I′} being the trivial representation of I′ Howe and Moy ([HM1]) proved that H(G//K_{d}, ρ) is isomorphic to H(G′//I′) in such

a way that the isomorphism preserves the natural inner products on the two algebras up to multiples and takes Plancherel measure to Plancherel measure. This Hecke algebra isomorphism induces a bijection between the set of (equivalence classes of) irreducible admissible representations π of G such that (K_{d}, ρ) ↪ π and the set of (equivalence classes of) irreducible admissible representations π′ of G′ such that (I′, 1_{I′}) ↪ π′. We will write π ↔ π′ for this correspondence.

Given O ∈ Ω_{G}(s_{d}), choose an element of O whose semisimple part is s_{d}. The nilpotent part of this element belongs to 𝔤′, hence defines an O′ ∈ Ω_{G′}(0). The map O ↦ O′ is a bijection, and we write O ↔ O′. Given a choice of G′-invariant measure on O′, we normalize the G-invariant measure on O so that

(∗)  μ_{O}(ch(𝔤l_{n}(O_{F}))) = |GL_{n}(𝔽)|_{p′}|GL_{n/d}(𝔼)|_{p′}^{−1} μ_{O′}(ch(𝔤l_{n/d}(O_{E}))).

Here | · |_{p′} denotes p′-cardinality (p the characteristic of 𝔽), and ch(S) the characteristic function of the set S. When evaluating Fourier transforms, we take B(X, Y) = trace(XY) and we assume that the character ψ has conductor 𝔭_{F}. Let 𝔤_{tn} be the set of topologically nilpotent elements in 𝔤. Our main result in this setting is:

Theorem. Suppose that π is irreducible and admissible and (K_{d}, ρ) ↪ π.

(1) There exist complex numbers c_{O}(π), O ∈ Ω_{G}(s_{d}), such that

Θ_{π}(1 + X) = ∑_{O∈Ω_{G}(s_{d})} c_{O}(π) μ̂_{O}(X) X ∈ 𝔤_{reg} ∩ 𝔤_{tn}.

(2) Let π′ be an irreducible admissible representation of G′ such that π ↔ π′. Given O′ ∈ Ω_{G′}(0), let c_{O′}(π′) denote the coefficient of μ_{O′} in the local character expansion of π′ at 1. If O ↔ O′ and the measures on O and O′ are related as above in (∗), then

c_{O}(π) = c_{O′}(π′), O ∈ Ω_{G}(s_{d}).

The orbits in Ω_{G}(s_{d}) can be parametrized by ordered partitions of n/d. Fix O ∈ Ω_{G}(s_{d}). Let O′ ∈ Ω_{G′}(0) be such that O ↔ O′. Choose a parabolic subalgebra P_{O′} of 𝔤′ such that the intersection of O′ with the nilradical of P_{O′} is open in that nilradical (that is, O′ is the Richardson class attached to P_{O′}). The Levi component of P_{O′} is isomorphic to 𝔤l_{α_{1}}(E) ⊕ ⋯ ⊕ 𝔤l_{α_{r}}(E) for some partition α = (α_{1}, …, α_{r}) of n/d. Up to re-ordering, α is determined by O′, and conversely. Set ℓ(α) = r and let |St(α)| be the cardinality of the stabilizer of α in the symmetric group on r letters.

Let P_{O′} be the parabolic subgroup of G′ having Lie algebra P_{O′}. We normalize μ_{O′} in such a way that μ_{O′} equals the germ of the character of Ind_{P_{O′}}^{G′} 1_{P_{O′}}. This then determines a normalization of μ_{O} via (∗).

Corollary. Let π be a generalized special (discrete series) representation of G such that (K_{d}, ρ) ↪ π. Given O ∈ Ω_{G}(s_{d}), define α as above.

c_{O}(π) = \frac{(−1)^{(n/d)−ℓ(α)}ℓ(α)!}{|St(α)|}, O ∈ Ω_{G}(s_{d})

Remarks:

(i) For the corollary, if π ↔ π′, then π′ = St_{G′}, the Steinberg (or special) representation of G′. As the character Θ_{St_{G′}} is known, (3) is obtained by substituting the value for c_{O′}(St_{G′}) into (2). In order to do that, it was necessary to specify a normalization for μ_{O′}. (ii) The condition (K_{d}, ρ) ↪ π guarantees the existence of non zero K_{d}^{u}-invariant vectors in the space of π. Denote the pro unipotent radical of K_{n} = GL_{n}(O_{F}) by K_{n}^{u}. Note that K_{n}^{u} ⊂ K_{d}^{u}. let V be the space of K_{n}^{u}-fixed vectors in the space of π. As K_{n}^{u} is normal in K_{n}, V is π | K_{n}-stable. Hence we get a representation κ of GL_{n}(𝔽) ≃ K_{n}/K_{n}^{u} on V. We assume that p is sufficiently large that results of Waldspurger ([W]) concerning uniformity of local character expansions can be applied. Then the local character expansion of π holds on 𝔤_{tn} ∩ 𝔤_{reg}, and the expansion is determined by the values of the character χ_{κ} of κ on the unipotent set U_{GL_{n}(𝔽)}.

Similarly, there exists a representation κ′ of GL_{n/d}(𝔼) whose character values on U_{GL_{n/d}(𝔼)} determine the local character expansion of π′ (which holds on 𝔤′_{tn} ∩ 𝔤′_{reg}).

The cuspidal representation σ of GL_{d}(𝔽) corresponds to a character θ of 𝔼^{×}, via Deligne-Lusztig induction: χ_{σ} = (−1)^{d−1} R𝔼^{×}^{GL_{d}(𝔽)}(θ). The Hecke algebra isomorphism of Howe and Moy restricts to an isomorphism of finite group Hecke algebras which we can use to show that χ_{κ} = (−1)^{n−(n/d)} R_{GL_{n/d}(𝔼)}^{GL_{n}(𝔽)}((θ ∘ det′) χ_{κ′}). Here, det′ is the determinant on

GL_{n/d}(𝔼) and R_{GL_{n/d}(𝔼)}^{GL_{n}(𝔽)} denotes Deligne-Lusztig induction ([L]). This relation between χ_{κ} and χ_{κ′} leads to relations between the germs of Θ_{π} and Θ_{π′}. These relations are obtained by evaluating Θ_{π} and Θ_{π′} on characteristic functions of pro unipotent radicals of parahoric subgroups: up to scalar multiples, this is equivalent to computing the dimension of the space of κ or κ′-fixed vectors for the unipotent radical of a parabolic subgroup of GL_{n}(𝔽), resp. GL_{n/d}(𝔼).

Let K be a parahoric subgroup of G with pro unipotent radical K^{u}. Let K be the corresponding parahoric O_{F} -subalgebra of 𝔤, with pro nilpotent radical K^{u}. In proving the theorem, we compare the values Θ_{π}(ch(K^{u})) and Θ_{π′}(ch(K^{′u}) for those parahoric subgroups K of G such that K′ = K ∩ G′ is a parahoric subgroup of G′. The descent formulas for orbital integrals which are useful for comparing the germs of Θ_{π} and Θ_{π′} are those relating μ_{O}(ch(K)) and μ_{O′}(ch(K′)) for parahoric subalgebras K and K′ of 𝔤 and 𝔤′ respectively, such that K′ = K ∩ 𝔤′. (iii) The relations between the values of χ_{κ} and χ_{κ′} on unipotent elements can be expressed in terms of Hall algebras of finite general linear groups. (iv) There is a finite collection S′ of regular semisimple unramified Ad G′ orbits having the property that, for any π′ such that (I′, 1_{I′}) ↪ π′, the germ of Θ_{π′} can be expressed as a linear combination of { μ_{O′} | O′ ∈ S′ }. Given X ∈ O′ ∈ S′, let O = O_{G}(s_{d} + X). Let S be the set of all such O. There is a result analogous to the above theorem which gives the germ of Θ_{π}, π ↔ π′, as a linear combination of { μ̂_{O} | O ∈ S }, including relations between the coefficients corresponding to the orbits O′ ∈ S′ and O ∈ S such that O′ ↔ O.

(v) Barbasch and Moy ([BM]) have given a recursive method for computing the coefficients in the local character expansions of depth zero irreducible admissible representations of reductive groups. The method uses generalized Gelfand-Graev representations of finite reductive groups. The method can be applied to calculate the coefficients for representations of groups of low rank (some examples are given in [BM]). In particular, Moy computed the values of the coefficients for a depth zero generalized special representation of GL_{4}(F) corresponding to d = 2 in our example. Using the corollary, after expressing each μ̂_{O}, O ∈ Ω_{GL_{4}(F)}(s_{2}) as a linear combination of μ̂_{O}, O ∈ Ω_{GL_{4}(F)}(0), we obtain the same result.

Example 2:

• G = GL_{n}(F). We assume p > n.

• d a positive integral divisor of n

• σ an irreducible supercuspidal representation of GL_{d}(F)

• π_{gs} the discrete series representation of G corresponding to σ

• E the degree d extension and θ the admissible quasi character of E^{×} which parametrizes the supercuspidal representation σ via Howe’s construction ([H])

• G′ = GL_{n/d}(E)

• (J, ρ) the Howe-Moy refined minimal K-type attached to π_{gs} ([CMS],[HM2])

• H(G//J, ρ) the ρ-spherical Hecke algebra, as for example 1 but with J replacing K_{d} The group J is an open compact subgroup of G having the property that I′ = J ∩G′ is

an Iwahori subgroup of G′. Howe and Moy ([HM2]) proved that H(G//J, ρ) is isomorphic to H(G′//I′) via an isomorphism which preserves support and is an L^{2} isometry. As for example 1, if (J, ρ) ↪ π and π corresponds via the Hecke algebra isomorphism to π′ such that (I′, 1_{I′}) ↪ π′, we write π ↔ π′.

There exists s_{σ} ∈ E ↪ 𝔤l_{d}(E) such that the germ of Θ_{σ} is given by d(σ) μ̂_{O_{GL_{d}(F)}(s_{σ})} ([Mu1]). Identifying Let s = (s_{σ}, …, s_{σ}) ∈ M_{d}(F) × ⋯ × M_{d}(F) ⊂ G. As s_{σ} generates E/F, it follows that G′ is isomorphic to the centralizer of s in G. There is a bijection between Ω_{G}(s) and Ω_{G′}(0) resulting from the Jordan decomposition. We expect that if μ_{O}, O ∈ Ω_{G}(s) and μ_{O′}, O′ ∈ Ω_{G′}(0) are suitably normalized, the analogue of the theorem and corollary in example 1 will hold in this more general setting. In particular, the germ of Θ_{π_{gs}} should be a linear combination of the μ̂_{O}, O ∈ Ω_{G}(s), with the coefficient of μ̂_{O} as in the corollary. Remarks:

(i) The Hecke algebra isomorphisms of Howe and Moy ([HM2]) relate admissible representations of GL_{n}(F) and unipotent representations of products of general linear groups. It appears that these relations tranlate into relations between the germs of characters of admissible representations and germs of characters of unipotent representations. It is natural to ask whether Hecke algebra isomorphisms for other reductive groups give rise to similar relations between germs of characters. (ii) In [CMS], Corwin, Moy and Sally used properties of the Hecke algebra isomorphisms discussed in example 2 to compute the formal degrees of all discrete series representations of GL_{n}(F).

Example 3:

• G = GSp_{4}(F); p odd

• E a quadratic unramified extension of F

• ε a root of unity in O_{E} whose image in 𝔼 generates 𝔼 over 𝔽; ε will also denote an image of this root of unity in GL_{2}(O_{F})

• M = GL_{2} × GL_{1}, viewed as a standard Levi subgroup of GSp_{4}, and M = M(F)

• s = (ε, 0) ∈ 𝔤l_{2}(F) ⊕ F = Lie(M); G′ ≃ { g = (x, α) ∈ GL_{2}(F) × E^{×} | det(g) = N_{E/F} (α) }, the stabilizer of s in G.

• K the parahoric subgroup of G which is equal to the inverse image in GSp_{4}(O_{F}) of a standard parabolic subgroup of GSp_{4}(𝔽) having Levi component M(𝔽), K^{u} the pro unipotent radical of K

• I′ = K ∩ G′, an Iwahori subgroup of G′

• σ an irreducible cuspidal representation of GL_{2}(𝔽), ρ the pullback to K of the representation σ ⊗ 1 of M(𝔽) ≃ K/K^{u}

In [My2], Moy classified the depth zero admissible representations of G via nondegenerate represetations and Hecke algebra isomorphisms. This example corresponds to case 3.62(i) of [My2]. Moy obtained an isomorphism of H(G//K, ρ) with the Iwahori spherical Hecke algebra of a group which he called G′, which differs from the one defined above. However, the Iwahori spherical Hecke algebra for Moy’s G′ coincides with the Iwahori spherical Hecke algebra for the above G′. Hence H(G//K, ρ) ≃ H(G′//I′). Preliminary calculations indicate that the analogues of the theorem and corollary obtained for our first example will hold in this setting. In particular, if π is the discrete series representation corresponding via the Hecke algebra isomorphism to the Steinberg representation St_{G′} of G′, the germ of Θ_{π} belongs to the span of { μ̂_{O}, O ∈ Ω_{G}(s) }, with the coefficients equal to the corresponding coefficients in the local character expansion for the germ of Θ_{St_{G′}}.

References

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Department of Mathematics University of Toronto Toronto, Canada M5S 3G3 E-mail address: fiona@math.toronto.edu