KAZHDAN/PATTERSON LIFTING FOR GL(n, ℝ)

Jeffrey Adams and Jing-Song Huang

§1. Introduction. Kazhdan and Patterson have conjectured the existence of a lifting theory, taking representations of GL(n, 𝔽) to representations of the nonlinear covering groups of GL(n, 𝔽). We consider the case 𝔽 = ℝ. We work directly on the level of global characters.

Let G = GL(n, ℝ) and p : G̃ → G its non-linear two-fold cover (see §2 for details). For s : G → G̃ a section, g → s(g)^{2} defines a map from conjugacy classes in G to conjugacy classes in G̃, independent of the choice of s. As in ([6],§2) we modify this by a certain function u : G → ker(p) to obtain t(g) = s(g)^{2}u(g).

Let π be a virtual module for G with global character θ_{π}, viewed as a function on the regular semisimple elements. For g a regular semisimple element in the image of t let

(1.1)  t_{∗}(θ_{π})(g) = 2^{[n/2]−n} ∑_{\{h∈G | t(h)=g\}} \frac{Δ(h)}{Δ(g)}Θ_{π}(h)

where Δ is the usual Weyl denominator. Let

(1.2)  Z_{0} = \{ \begin{matrix}I & n even \\ ±I & n odd \end{matrix} , Z̃_{0} = p^{−1}(Z_{0});

this is a central subgroup of G̃. Fix a genuine character χ_{0} of Z̃_{0}; there are two such choices if n is odd, and one if n is even. For g ∈ t(G), z ∈ Z̃_{0} define

(1.3)  t_{∗}(θ_{π})(gz) = t_{∗}(Θ_{π})(g)χ_{0}(z).

Finally set t_{∗}(Θ_{π})(g) = 0 if g is not in the set t(G)Z̃_{0}.

The conjecture of Kazhdan and Patterson ([6], end of §4) says that if π is irreducible, then t_{∗}(π) is an irreducible representation, up to sign, or zero. Our main result is:

Theorem 1.4. Let π be an irreducible unitary representation of G. Then t_{∗}(π) is either zero, or an irreducible unitary representation, up to sign. It may be computed explicitly (Proposition 5.4).

A basic property of t_{∗} is that it commutes with real parabolic induction. Together with the fact [2] that for GL(2) it takes the discrete series to the discrete series, it

Typeset by AMS-TEX

The first author was supported by NSF grant DMS-94-01074. The second author was partially supported by NSF Grant DMS 9306138 and RGC Competitive Earmarked Research Grant HKUST 588/94P.

follows that t_{∗} takes virtual modules for G to genuine virtual modules for G̃. By the classification of the unitary dual of G [16] to compute t_{∗} it suffices to compute t_{∗}(π) for a small number of basic cases: the trivial and sgn representations, and the Speh representations. Write Speh(k) for the k^{th} Speh representation (cf. §5) of G̃L(2n, ℝ). Here k = \frac{1}{2}, 1, \frac{3}{2}, …, and Speh(k) factors to GL(2n, ℝ) if k is an integer.

Recall [3] G̃ has a distinguished irreducible unitary representation T_{n} if n is even. If n is odd G̃ has two such representations T_{n}(χ_{0}), depending on the choice of χ_{0} (Z̃_{0} acts in T_{n}(χ_{0}) by χ_{0}.) These are “almost-spherical” representations with infinitesimal character ρ/2.

Proposition 1.5.

(1) If n is even, t_{∗}(11) = T_{n}, and t_{∗}(sgn) = Speh(1/2).

(2) If n is odd t_{∗}(11) = T_{n}(χ_{0}), and t_{∗}(sgn) = 0.

Let 𝔮 = 𝔩 ⊕ 𝔲 be a parabolic subalgebra of 𝔤𝔩(2n, ℂ) and write R_{𝔮} for the corresponding derived functor in the middle degree. This takes representations of GL(n, ℂ) to representations of GL(2n, ℝ). The Speh representations are realized as Speh(k) = R_{𝔮}(χ(k)) where χ(k) = (det/|det|)^{k}. Then

(1.6)  t_{∗}R_{𝔮}(χ(k)) = R_{𝔮}(t_{∗}^{L}(χ(k))

where t_{∗}^{L} is (up to twist — cf. (3.8)) Kazhdan-Patterson lifting for GL(n, ℂ) as studied in [11]. Then t_{∗}^{L}(χ(k)) may be read off from [11], and the value of t_{∗}Speh(k) deduced from this (Proposition 5.4).

Proposition 1.5 and (1.6) follow easily from two properties of t_{∗}:

(1) t_{∗} commutes with coherent continuation,

(2) t_{∗} commutes with the Euler character of cohomological induction.

A bit more precisely, if {π(λ + μ)} (as μ runs over the weights of a Cartan subgroup) is a coherent family of virtual G-modules, the same holds for {t_{∗}π(λ + μ)}, with a slight twist if n is odd (Proposition 4.3). Secondly, suppose L is a θ-stable Levi factor of a parabolic subalgebra 𝔮 of 𝔤𝔩(n, ℂ); L is a product of factors GL(m_{i}, ℝ) and GL(m_{i}, ℂ). Write t_{∗}^{L} for Kazhdan-Patterson lifting for L (modified on GL(m_{i}, ℂ) factors by (3.8)). Let R̂_{𝔮} = ∑_{i} R_{𝔮}^{i}. Then for any L-module π:

(1.7)  t_{∗}R̂_{𝔮}(π) = R̂_{𝔮}(t_{∗}^{L}(π)).

The map t_{∗} from irreducible unitary representations of G to those of G̃ is not surjective. By considering complementary series it is easy to find examples of irreducible unitary representations π̃ of G̃, for which there is a representation π of G satisfying t_{∗}(π) = π̃, but π is either not irreducible, or not unitary (cf. Remark 5.9). The problem of extending the main result to all irreducible representations is non-trivial.

We thank Heng Sun, David Kazhdan and David Vogan for helpful discussions.

§2. Lifting: Definition and Basic Properties.

Let G = GL(n, ℝ) and G̃ its two-fold cover as defined in [8], cf. [5]. This is realized as

{(g, ϵ) | g ∈ G, ϵ = ±1}

with multiplication given by a cocycle c(g, g′), i.e. (g, ϵ)(g′, ϵ′) = (gg′, ϵϵ′c(g, g′)).

As noticed by a number of people the formula for the cocycle in [5] is not correct over a field 𝔽 for which (−1, −1) = −1 (( , ) is the Hilbert symbol). This is the case over ℝ. In this case the cocycle is not trivial on the permutation matrices W ≃ S^{n}; this covering is obtained by restricting the P in cover of O(n), and is non-trivial. It is difficult to give an explicit formula for this cocycle; it is non-trivial on A^{n} and is not symmetric. In addition the final formula on page 39 of [5] requires an extra term of σ(w_{1}, w_{2}).

For g ∈ G let

(2.1)(a)  t(g) = s(g)^{2}c(g, g)

= (g^{2}, 1) (2.1)(b)

where s : G → G̃ is any section. This is the definition of ([6],§4) specialized to ℝ and the covering of order 2, since in this case u(g) = c(g, g) (in the notation of [6]).

For H a subgroup of G, H̃ will always denote the inverse image of H in G̃. Let Z be the center of G, and \overline{Z} ⊂ Z̃ the center of G̃. Then \overline{Z} = \{(xI, ϵ) | x^{n−1} > 0\}. Furthermore t : Z → \overline{Z} is a homomorphism, and with Z̃_{0} as in (1.2):

(2.2)  \overline{Z} = t(Z)Z̃_{0} (direct product).

Fix a genuine character χ_{0} of Z̃_{0}. For χ a character of Z let χ̃ be the genuine character of \overline{Z} defined by

(2.3)  χ̃(t(z)w) = χ(z)χ_{0}(w) (z ∈ Z, w ∈ Z̃_{0}).

For g a semisimple element of G let Δ(g) be the (absolute value of the) usual Weyl denominator, i.e. ∏_{i<j} |z_{i} − z_{j}|/|z_{i}z_{j}|^{\frac{1}{2}} where z_{1}, …, z_{n} are the eigenvalues of g.

Following ([6], Proposition 5.6) and ([4], 26.1), we define lifting of characters as follows. Let

(2.4)(a)  G_{0} = {g^{2}z | g^{2} ∈ G regular semisimple, z ∈ Z_{0}}, G̃_{0} = p^{−1}(G_{0}).

In fact every element g of G̃_{0} may be written uniquely in the form

(2.4)(b)  g = g′z (g′ ∈ t(G), z ∈ Z̃_{0}).

As in ([2], Lemma 1.1.1) if g is a regular semisimple element of G̃ not in G̃_{0} then the character of any genuine virtual representation vanishes on g, since g is conjugate to ζg for ζ the non-trivial element in p^{−1}(1).

Definition 2.5. For π a virtual module for G and g a regular semisimple element of G̃, let

t_{∗}(θ_{π})(g) = 2^{[n/2]−n} \begin{aligned}& ∑ \\ \{ & \begin{matrix}h ∈ G, z ∈ Z̃_{0} \\ t(h)z = g \end{matrix} & \} \end{aligned} \frac{Δ(h)}{Δ(g)}Θ_{π}(h)χ_{0}(z).

Thus t_{∗}(θ_{π}) is a conjugation-invariant function on the regular semisimple elements of G̃. In fact (Corollary 3.17) it is (the function representing) a virtual character. The equivalence of this definition and that of [4] follows from (2.4)(b).

Note that

t_{∗}(θ_{π})(gz) = t_{∗}(θ_{π})(g)χ_{0}(z) (g ∈ t(G), z ∈ Z̃_{0}),

so t_{∗}(θ_{π}) is determined by χ_{0} and its restriction to t(G).

Suppose π has central character χ_{π}. Then

(2.6)(a)  t_{∗}(π) = 0 if χ_{π}(−I) = −1.

On the other hand if χ_{π}(−I) = 1 then by (2.3)

(2.6)(b)  t_{∗}(θ_{π})(gz) = t_{∗}(Θ_{π})(g)χ̃_{π}(z) (g ∈ G̃, z ∈ Z̃).

The definition implies

(2.7)  t_{∗}(π|det|^{ν}) = t_{∗}(π)|det|^{ν/2} for all ν ∈ ℂ.

Suppose π is irreducible with central character χ_{π}. If n is odd, precisely one of π and π ⊗ sgn satisfy χ_{π}(−I) = 1. If n is even either both of π and π ⊗ sgn satisfy this or neither do; the relationship between t_{∗}(π) and t_{∗}(π ⊗ sgn) is subtle (as is seen already in Proposition 1.5).

If n is odd the two choices of t_{∗}^{+} and t_{∗}^{−} of lifting, corresponding to the choice of χ_{0}, satisfy t_{∗}^{−}(π) = t_{∗}^{+}(π) ⊗ sgn for all π.

Remark 2.8. A new cover G̃′ of G is obtained by taking the cocycle

c′(g, g′) = c(g, g′)(det(g), det(g′))_{ℝ}

where ( , )_{ℝ} is the Hilbert symbol. There is a bijection π → π′ between the genuine irreducible representations of G̃ and those of G̃′, given by

π′(g, ϵ) = π(g, ϵ)γ(det(g)).

Here γ is the Weil index ([9],appendix) of ℝ determined by a non-trivial additive character of ℝ; this satisfies

γ(x)γ(y) = γ(xy)(x, y)_{ℝ} (x, y ∈ ℝ^{∗}).

§3. Induction.

A basic property of t_{∗} (over any field) is that commutes with parabolic induction ([4], Proposition 26.2). In the real case an important tool of [16] is cohomological induction. The main result of this section is that t_{∗} commutes with these functors as well, provided one takes the Euler character. In particular this holds for cohomological induction from GL(n, ℂ) to GL(2n, ℝ).

We begin with a brief discussion of real parabolic induction. We follow ([4],§26.2), adapted to the real case.

Let P = MN be a parabolic subgroup of G. We identify Ñ with N as usual and write P̃ = M̃N. Write M = M_{1} × ⋯ × M_{r} with M_{i} ≃ GL(m_{i}, ℝ). Suppose σ_{i} (i ≤ i ≤ r) is a genuine representation of M̃_{i}. Let p (resp. q) be the number of factors for which m_{i} is even (resp. odd). Since the M̃_{i} do not commute, some care is necessary in defining induction.

Let M_{i}^{+} (resp. M^{+}) be the identity component of M_{i} (resp. M). Then M̃_{i}^{+}, M̃_{j}^{+} commute in G̃ for i ≠ j. With χ_{0} a fixed character of Z̃_{0} form the representation σ_{1} ⊗ ⋯ ⊗ σ_{r} ⊗ χ_{0} of M̃_{1} × ⋯ × M̃_{r} × Z̃_{0}. The restriction to M̃_{1}^{+} × ⋯ × M̃_{r}^{+} × Z̃_{0} decomposes into the direct sum of 2^{p} non-isomorphic summands; let σ be any such summand. This factors to a representation, also denoted σ, of M̃^{+}Z̃_{0}. Then

(3.1)  IndM̃ ; M̃^{+}Z̃_{0}(σ)

is independent of the choice of σ, and is the direct sum of 2^{[q/2]} copies of an irreducible representation τ. This can be proved directly, or follows from ([3], Definition 2.5) applied repeatedly, and induction in stages.

We define

(3.2)(a)  IndG̃ ; M̃N(σ_{1} ⊗ ⋯ ⊗ σ_{r})

to be

(3.2)(b)  2^{−[q/2]}IndG̃ ; M̃^{+}Z̃_{0}N(σ) = IndG̃ ; M̃N(τ).

If n is odd this depends on the choice of χ_{0}. This satisfies induction in stages.

Applying Definition 2.5 to each M_{i}, we obtain t_{∗}^{M}, taking virtual modules for M = M_{1} × ⋯ × M_{r} to genuine virtual modules for M̃_{1} × ⋯ × M̃_{r}.

Lemma 3.3 [4, Proposition 26.2]. Suppose σ_{1} ⊗ ⋯ ⊗ σ_{r} is a virtual module for M = M_{1} × ⋯ × M_{r}. Then

(3.4)  IndG̃ ; M̃N(t_{∗}^{M}(σ_{1} ⊗ ⋯ ⊗ σ_{r})) = t_{∗}Ind_{MN}^{G}(σ_{1} ⊗ ⋯ ⊗ σ_{r}).

3.5 Remark. This follows from the induced character formula as in [2], [4]. A comment is in order regarding the constants. The value of the character of the right hand side of (3.4) at a regular semisimple element has a coefficient of 2^{[n/2]−n} coming from Definition 2.5. The corresponding coefficient on the left hand side is

(3.6)  (∏_{i=1}^{r} 2^{[m_{i}/2]−m_{i}}) 2^{−[q/2]}|M̃ : M̃^{+}Z̃_{0}|2^{−p}.

These terms are from Definition 2.5, (3.2)(b), induction from M̃^{+}Z̃_{0} to M̃, and restriction from M̃_{1} × … to M̃_{1}^{+} × … respectively. Noting that |M̃ : M̃^{+}Z̃_{0}| = 2^{r} (if n is even) or 2^{r−1} (if n is odd), a simple computation shows (3.6) equals 2^{[n/2]−n}.

We now consider general induction functors. We recall some definitions from ([16],§17). Fix a Cartan involution θ(g) = ^{t}g^{−1} of G, and K = G^{θ} = O(n). Let L be an arbitrary θ-stable Levi subgroup of G, i.e. L is the centralizer of a θ-stable reductive abelian subalgebra of 𝔤. Let 𝔮 = 𝔩 ⊕ 𝔲 be a parabolic subalgebra of 𝔤. Define the functors R_{𝔮}^{i}, from (𝔩, L ∩ K) to (𝔤, K) modules, normalized as in ([7], 11.150b). For brevity we write L-module for (𝔩, L ∩ K)-module, and similarly for G, L̃, etc. As in ([16], Definition 17.1) the cover of L is not necessary and τ̃ can be chosen to be a character of L. The normalization is such that R_{𝔮}^{i} preserves infinitesimal character.

Let R̂_{𝔮} be the Euler character of R_{𝔮}^{∗}:

(3.7)  R̂_{𝔮} = (−1)^{S} ∑_{i}(−1)^{i}R_{𝔮}^{i}

with S = dim(𝔲 ∩ 𝔨).

With L̃ the inverse image of L in G̃ as usual, R_{𝔮}^{i} and R̂_{𝔮} are defined, taking L̃ modules to G̃ modules. Write

(3.8)(a)  L = L_{1} × ⋯ × L_{r}

with

(3.8)(b)  L_{i} ≃ GL(m_{i}, ℂ) or GL(m_{i}, ℝ) (1 ≤ i ≤ r).

The inverse images of the GL(m_{i}, ℂ) terms commute in L̃. For σ_{i} a genuine representation of L_{i} (1 ≤ i ≤ r) we define R_{𝔮}^{∗}(σ_{1} ⊗ ⋯ ⊗ σ_{r}), with the preceding considerations applied to the GL(n_{i}, ℝ) terms.

When 𝔮 is understood we suppress it from the notation and write R̂_{L}^{G} = R̂_{𝔮} and R̂G̃ ; L̃ = R̂_{𝔮} for the functors for G and G̃ respectively.

We now define lifting t_{∗}^{L}, taking virtual modules for L to genuine virtual modules for L̃. If L_{i} ≃ GL(m_{i}, ℝ) then the definitions of §2 apply to define t_{∗} for L_{i}. Suppose L_{i} ≃ GL(m_{i}, ℂ). Let t_{∗}^{0} denote the lifting of ([11], [6]) applied to L_{i}. (The map on conjugacy classes is t(g) = g^{2}, i.e. n = 2 in the notation of [11].) This takes the space of virtual modules for L_{i} to itself.

Choose an isomorphism ϕ : L_{i} → GL(m_{i}, ℂ). Then L̃_{i} has a unique character χ satisfying

(3.9)(a)  χ^{2} = (det/|det|) ∘ ϕ.

For π a virtual module for L_{i} ≃ GL(m_{i}, ℂ) let

(3.9)(b)  t_{∗}(π) = χ ⊗ t_{∗}^{0}(π ⊗ χ^{−2}).

This is a genuine virtual module for L̃_{i}, and is independent of the choice of ϕ.

Finally taking the product over i defines t_{∗}^{L}, taking virtual modules for L = L_{1} × ⋯ × L_{r} to genuine virtual modules for L̃ = L̃_{1} × ⋯ × L̃_{r}.

Suppose L_{i} = GL(m_{i}, ℂ). Let X(λ_{1}, λ_{2}) be a standard module for L_{i} in the usual notation [17]. This notation extends in the obvious way to representations of L̃_{i}. Here λ_{i} ∈ ℂ^{m_{i}} and μ = λ_{1} − λ_{2} ∈ ℤ^{m_{i}} for L_{i} or (ℤ + \frac{1}{2})^{m_{i}} for genuine representations of L̃_{i}. Then restricted to this factor,

(3.10)(a)  t_{∗}^{L}X(λ_{1}, λ_{2}) = \{ \begin{matrix}X(λ_{1}/2, λ_{2}/2) & μ = λ_{1} − λ_{2} ∈ (2ℤ + 1)^{n} \\ 0 & otherwise. \end{matrix}

For later use we write this as

(3.10)(b)  t_{∗}^{L}R̂_{T}^{L}(π) = RL̃ ; T̃(t_{∗}^{T} (π))

Here T = ℂ^{∗m_{i}}, π is a character of T, and R̂_{T}^{L} = R_{𝔟}^{0} is ordinary parabolic induction from a Borel subgroup B containing T.

Theorem 3.11. Fix 𝔮 = 𝔩 ⊕ 𝔲, L and a virtual (𝔩, L ∩ K)-module π. Then

t_{∗}R̂_{L}^{G}(π) = R̂G̃ ; L̃(t_{∗}^{L}(π)).

Before giving the proof we state a few lemmas. Suppose L ⊂ L′ ⊂ G are Levi subgroups, satisfying L∩K ⊂ L′ ∩K. Suppose 𝔮′ = 𝔩′ ⊕𝔲′ is a parabolic subalgebra of 𝔤, and 𝔮 = 𝔩 ⊕ 𝔲 is a parabolic subalgebra of 𝔩′. Let 𝔮′′ = 𝔩 ⊕ 𝔲′ ⊕ 𝔲. Then R̂_{𝔮′′} and R̂_{𝔮′} are defined, taking L, L′ modules to G-modules, respectively, and R_{𝔮} takes L-modules to L′-modules. (Here we mean (𝔩, L ∩ K)-modules, etc.) As above write R̂_{L}^{L′} = R̂_{𝔮}, R̂_{L′}^{G} = R̂_{𝔮′}, and R̂_{L}^{G} = R̂_{𝔮′′}.

These functors are related by a spectral sequence ([7], Theorem 11.77, or [14], Proposition 6.3.6). The Euler character χ (with respect to total degree) of this spectral sequence E_{k} satisfies χ(E_{2}) = χ(E_{3}) = ⋯ = χ(E_{∞}). The next Lemma is an immediate consequence.

Lemma 3.12 (Induction in Stages). For any L-module π:

R̂_{L}^{G}(π) ≃ R̂_{L′}^{G}(R̂_{L}^{L′}(π)).

We also need a strong independence of polarization result for R̂_{𝔮}. For 𝔥 a Cartan subalgebra of 𝔤 with Δ^{+} a set of positive roots, write Δ_{im}^{+}, Δ_{re}^{+}, Δ_{cx}^{+} for the imaginary, real, and complex roots of Δ^{+}, respectively.

Lemma 3.13 (Independence of Polarization). Fix a Cartan subgroup H of G or G̃ and and two Borel subalgebras 𝔮_{1}, 𝔮_{2} containing 𝔥. Write Δ_{1}^{+}, Δ_{2}^{+} for the roots of 𝔥 in 𝔮_{1}, 𝔮_{2} respectively. Assume

(3.14)  Δ_{1,re}^{+} = Δ_{2,re}^{+}, Δ_{1,im}^{+} = Δ_{2,im}^{+}.

Then for any (𝔥, H ∩ K)-module π:

R̂_{𝔮_{2}}(π) = R̂_{𝔮_{1}}(π)

(equality in the Grothendieck group).

Proof. We may assume π is irreducible, and 𝔥 acts in π by an element λ ∈ 𝔥^{∗}. (In the case of G, H is abelian and λ = dπ.) For λ regular the result follows by repeated application of ([7], Theorem 11.87). Apply (a) or (b) of that result; taking the Euler character the result follows by ([7], Theorem 0.46b). The twist by ℂ_{α} isn’t needed since we are using normalized induction. The general case follows by coherent continuation: both sides of the statement form coherent families by ([14], Corollary 7.2.10), which agree for regular λ, and therefore they agree for all λ.

Proof of Theorem 3.11. case 1: G = GL(2, ℝ). This is covered by [2]. More precisely this is the case for L the split Cartan subgroup. If L = T ≃ ℂ^{∗}, the fundamental Cartan subgroup, write λ = dπ ∈ 𝔱^{∗} and R̂_{L}^{G}(λ) = R̂_{L}^{G}(π). If (α, λ) > 0 for α the root of 𝔱 in 𝔮, then R̂_{L}^{G}(λ) is a relative discrete series representation and this case is also treated in [2]. The same calculation treats the general case (α, λ) ≤ 0. That is R̂_{L}^{G}(λ) has a character formula which is uniform in λ, as does R̂G̃ ; L̃(λ), and the same calculation for (α, λ) > 0 applies in general. (More precisely, both sides of the assertion form coherent families in λ, by Proposition 4.3, which can be proved directly for GL(2, ℝ). Since they agree for (α, λ) > 0, they agree for all λ.)

This case can also be proved by direct calculation using known character formulas for GL(2, ℝ) and G̃L(2, ℝ). case 2: L = ℂ^{∗p} × ℝ^{∗q}. Let L′ = GL(2, ℝ)^{p} × ℝ^{∗q} be the centralizer of the split component ℝ^{∗(p+q)} of L. By Lemma 3.13 we are free to replace 𝔮 by a parabolic subalgebra satisfying (3.14). After such a change we may assume there is a parabolic subalgebra 𝔮′ = 𝔩′ ⊕ 𝔲′ such that 𝔮 = 𝔩 ⊕ (𝔩′ ∩ 𝔲) ⊕ 𝔲′. That is we are in the setting of Lemma 3.12, and we compute:

t_{∗}R̂_{L}^{G}(π) = t_{∗}R̂_{L′}^{G} R̂_{L}^{L′}(π) (Lemma 3.12)

= R̂G̃ ; L̃′t_{∗}^{L′} R̂_{L}^{L′}(π) (Lemma 3.3) (3.15)

= R̂G̃ ; L̃′ R̂L̃′ ; L̃(t_{∗}^{L}(π)) (case 1)

= R̂G̃ ; L̃(t_{∗}^{L}(π)) (Lemma 3.12).

case 3: General L. It is enough to prove the result for π a standard module, i.e. π = R_{𝔮′}^{S}(σ), where T is a Cartan subgroup of L, 𝔮′ = 𝔱⊕𝔲′ is a parabolic subalgebra of 𝔩 and S = dim(𝔲 ∩ 𝔨). The data can be chosen so that R_{𝔮′}^{i}(σ) = 0 for i ≠ S, so π = R̂_{T}^{L}(σ). Let 𝔮′′ = 𝔱 ⊕ 𝔲′ ⊕ 𝔲. Then

t_{∗}R̂_{L}^{G}(π) = t_{∗}R̂_{L}^{G}R̂_{T}^{L}(σ)
= t_{∗}R̂_{T}^{G}(σ) (Lemma 3.12)

= R̂G̃ ; T̃(t_{∗}^{T} (σ)) (case 2) (3.16)

= R̂G̃ ; L̃R̂L̃ ; T̃(t_{∗}^{T} (σ)) (Lemma 3.12)

= R̂G̃ ; L̃t_{∗}^{L}R̂_{T}^{L}(σ) (case 2 and 3.10)

= R̂G̃ ; L̃t_{∗}^{L}(π).

This completes the proof.

Corollary 3.17.

(1) For any virtual module π, t_{∗}(π) is a virtual module.

(2) If π has infinitesimal character λ then t_{∗}(π) has infinitesimal character λ/2 (or is zero)

(We are identifying an infinitesimal character with an element of the dual of a Cartan subalgebra as usual.)

Proof. This holds for standard modules by the Theorem. Since these span the virtual modules the result follows.

Theorem 3.11 is most useful when the functors vanish except in one degree S, in which case the statement becomes

(3.18)  t_{∗}R_{𝔮}^{S}(π) = R_{𝔮}^{S}(t_{∗}^{L}(π)).

This is the case in our main application: L = GL(n, ℂ) and π is a one-dimensional representation of L. We consider this next.

For k ∈ ℤ and ν ∈ ℂ define a character of ℂ^{∗} by

(3.19)  χ(k, ν)(z) = (z/|z|)^{k}|z|^{ν} (z ∈ ℂ^{∗}).

We use the same notation for the restriction of this character to ℝ^{∗} (we may take k = 0, 1), as well as for the characters of GL(n, ℂ) and GL(n, ℝ) obtained by composing with det. Finally we use the same notation for characters of G̃L(n, ℝ) and G̃L(n, ℂ) obtained by composing with p. Let χ(k) = χ(k, 0).

Fix a θ-stable parabolic subalgebra 𝔮 = 𝔩 ⊕ 𝔲 of GL(2n, ℝ) with L ≃ GL(n, ℂ). Let 𝔠 be the center of 𝔩. Choose an isomorphism ϕ : L → GL(n, ℂ) and let χ(k, ν) = ϕ ∘ χ(k, ν). By replacing ϕ with ^{t} ϕ^{−1} if necessary we may assume k > 0 implies (α, dχ(k)) > 0 for all roots α of 𝔠 in 𝔲.

Let S = n(n−1)/2. Then for k a non-negative integer R_{𝔮}^{S}(χ(k)) is an irreducible unitary representation (the k^{th} Speh representation - see §5), and R_{𝔮}^{i}(χ(k)) = 0 for i ≠ S. By Theorem 3.11 we conclude:

(3.20)  t_{∗}R_{𝔮}^{S}(χ(k)) = R̂_{𝔮}(t_{∗}^{L}(χ(k))).

After taking (3.8) into account, t_{∗}^{L}(χ(k)) may be read off from [11]. The result is the following.

Lemma 3.21 [11].

(1) L = GL(2n, ℂ), k ≡ 0 mod 2:

t_{∗}^{L}(χ(k)) = (−1)^{n}IndL̃ ; MN(σ) with \{ \begin{matrix}M & = G̃L(n, ℂ) × G̃L(n, ℂ) \\ σ & = χ(\frac{k+1}{2}) ⊗ χ(\frac{k−1}{2}). \end{matrix}

(2) L = GL(2n, ℂ), k ≡ 1 mod 2:

t_{∗}^{L}(χ(k)) = IndL̃ ; MN(σ) with \{ \begin{matrix}M & = G̃L(n, ℂ) × G̃L(n, ℂ) \\ σ & = χ(\frac{k}{2} , \frac{1}{2}) ⊗ χ(\frac{k}{2} , −\frac{1}{2}). \end{matrix}

(3) L = GL(2n + 1, ℂ), k ≡ 1 mod 2:

t_{∗}^{L}(χ(k)) = (−1)^{n}IndL̃ ; MN(σ) with \{ \begin{matrix}M & = G̃L(n + 1, ℂ) × G̃L(n, ℂ) \\ σ & = χ(\frac{k}{2} ) ⊗ χ(\frac{k}{2} ). \end{matrix}

(4) L = GL(2n + 1, ℂ), k ≡ 0 mod 2: t_{∗}^{L}(χ(k)) = 0.

We reiterate this differs from [11] by the twist (3.8).

Lemma 3.22. For k = 0, 1, 2, …

R_{𝔮}^{S}(χ(k)) = R_{𝔮}^{S}(t_{∗}^{L}(χ(k))).

Furthermore t_{∗}R_{𝔮}^{S}(χ(k)) is an irreducible unitary representation, up to sign, or zero.

Proof. In each case t_{∗}^{L}(χ(k)) is an irreducible, unitary “basic almost spherical” representation in the sense of ([3], Definition 5.4). The Lemma follows from ([3], Proposition 7.1 and the proof of Theorem 6.7).

Combining the preceding two results gives an explicit description of t_{∗}R_{𝔮}^{S}(χ(k)) in terms of R_{𝔮}^{S}. In section 5 we express this in terms of real parabolic induction (Proposition 5.4).

§4. Coherent Continuation.

The main result of this section (Proposition 4.3) is that t_{∗} commutes with coherent continuation. This can also be deduced from Theorem 3.11.

Let T be the diagonal subgroup of G, or its inverse image in G̃. By the weight lattice Λ of T we mean the subgroup of T , written additively, consisting of weights of finite dimensional representations of G. (These are pulled back to G̃ in the case of the cover. If n > 1 every finite dimensional representation of G̃ factors to G, but for n = 1 this is not the case.)

Fix λ ∈ 𝔱^{∗}. Recall ([14], Definition 7.2.5) a coherent family based at λ is a collection of virtual modules

(4.1)(a)  {π(λ + μ) | μ ∈ Λ}

such that for all μ ∈ Λ

(4.1)(b)  π(λ + μ) has infinitesimal character λ + dμ

and for any finite dimensional representation F of G

(4.1)(c)  π(λ + μ) ⊗ F ≃ ∑_{γ∈Δ(F)} π(λ + (μ + γ)).

Here Δ(F) ⊂ Λ is the weights of F.

Fix λ and a coherent family {π(λ + μ) | μ ∈ Λ} for G. For μ ∈ Λ define

(4.2) π̃(\frac{λ}{2} + μ) = \{ \begin{matrix}t_{∗}π(λ + 2μ) & n even, or n odd and μ(−I) = 1 \\ [t_{∗}π(λ + 2μ)] ⊗ sgn & n odd, μ(−I) = −1 \end{matrix}

The sgn term is to correct the central character.

Proposition 4.3. \{π̃(\frac{λ}{2} + μ) | μ ∈ Λ\} is a coherent family based at \frac{λ}{2}.

Before giving the proof we recall the Adams^{1} operator.

Lemma 4.4. Let F be a finite dimensional representation of G = GL(n, ℝ). Then there exists a virtual finite dimensional representation A ∗ F (i.e. a finite integral linear combination of finite dimensional representations) satisfying

(4.5)(a)  Θ_{A∗F} (g) = Θ_{F} (g^{2})

for all g in G. Equivalently for all Cartan subgroups T of G

(4.5)(b)  Δ(A ∗ F) = 2Δ(F)

where Δ(F) ⊂ T denotes the weights of T.

For SL(n, ℝ), or equivalently SU(n), see ([1], §II.7). There is no difficulty extending this result to GL(n, ℝ).

Proof of Proposition 4.3. By Corollary 3.17, π̃(\frac{λ}{2} + μ) is a virtual character with infinitesimal character \frac{λ}{2} + dμ. We need to show for any finite dimensional representation F and μ ∈ Λ we have

(4.6)  ∑_{γ∈Δ(F)} π̃(\frac{λ}{2} + μ + γ) = π̃(\frac{λ}{2} + μ) ⊗ F

We compute the character of both sides at a regular semisimple element g ∈ G̃.

First suppose g ∈ t(G). The following sums over h are over {h ∈ G | t(h) = g}, and we write c = 2^{[n/2]−n}.

∑_{γ∈Δ(F)} Θ_{π̃( \frac{λ}{2} +μ+γ)}(g) = c ∑_{γ∈Δ(F)} ∑_{h} \frac{Δ(h)}{Δ(g)}Θ_{π(λ+2μ+2γ)}(h) (by Definition 2.5)

= c ∑_{h} ∑_{γ∈Δ(A∗F)} \frac{Δ(h)}{Δ(g)}Θ_{π(λ+2μ+γ)}(h) ((4.5)(b))

= c ∑_{h} \frac{Δ(h)}{Δ(g)}Θ_{π(λ+2μ)}(h)Θ_{A∗F} (h) ((4.1)(c)) (4.7)

= c ∑_{h} \frac{Δ(h)}{Δ(g)}Θ_{π(λ+2μ)}(h)Θ_{F} (g) ((4.5)(a))

= Θ_{π̃( \frac{λ}{2} +μ)}(g)Θ_{F} (g) (Definition 2.5).

It remains to check (4.6) on elements gz (g ∈ t(G), z ∈ Z̃_{0}) (cf. (2.4)(b)). Since both sides of (4.6) are genuine this is immediate if n is even. If n is odd it follows from the second case in (4.2). This completes the proof.

Fix a Cartan subalgebra 𝔥 of 𝔤. Let Δ(\frac{ρ}{2}) be the integral roots defined by \frac{ρ}{2} ([14], Definition 7.2.16) (here ρ is half the sum of any choice of positive roots.) Fix a virtual module π of G̃ with infinitesimal character \frac{ρ}{2}. We say π has maximal τ-invariant if the τ-invariant of π ([13], Definition 3.3) is equal to all simple roots (for any choice of positive roots Δ^{+}(\frac{ρ}{2})). Equivalently the translation of π to any integral root wall is zero. (See [15] for a further discussion of this definition.)

1No relation to either author

Corollary 4.7. Let π be a one-dimensional representation of G, and let π̃ = t_{∗}(π). Assume π̃ ≠ 0. Then π̃ has infinitesimal character ρ/2 and maximal τ-invariant.

Proof. The first assertion is Corollary 3.17. Suppose λ is singular and \frac{ρ}{2} − λ is a sum of roots. Since π is finite dimensional the translation of π to infinitesimal character 2λ is zero; by the Proposition the same holds for the translation of π̃ to λ.

The Corollary severely limits the irreducible representations π̃ which can contribute to the lift of a one-dimensional representation: π̃ must have infinitesimal character \frac{ρ}{2} and maximal τ-invariant. A simple generalization of the argument at the end of the proof of Lemma 11.4 in [3] shows that the only possibilities for π are:

(1) T_{n} and Speh(1/2) if n is even,

(2) T_{n}(χ_{0}) if n is odd.

Therefore for some constants a_{n}, a′_{n}, b_{n}, b′_{n}:

(4.8)  t_{∗}(11) = \{ \begin{matrix}a_{n}T_{n} + a′_{n}Speh(1/2) & n even \\ a_{n}T_{n}(χ_{0}) & n odd \end{matrix}
t_{∗}(sgn) = \{ \begin{matrix}b_{n}T_{n} + b′_{n}Speh(1/2) & n even \\ 0 & n odd \end{matrix}

(t_{∗}(sgn)) = 0 if n is odd by (2.6)(a)). To prove Proposition 1.5 we need to show a_{n} = 1, a′_{n} = b_{n} = 0 and b′_{n} = 1. We obtain some elementary information here, and defer the final result to the next section.

Let X be the standard module for G which contains the trivial representation as a constituent. Thus X = Ind_{T N}^{G}(τ) where T = ℝ^{∗n}, τ is trivial on (ℤ/2ℤ)^{n} and dτ = ρ. It follows immediately from Lemma 3.3 and ([3], §4) that t_{∗}(X) = X̃, where X̃ is the standard module for G̃ containing T_{n} as a constituent (of multiplicity one). Write X̃ = T_{n} + Z (in the Grothendieck group).

Write 11 as a linear combination of standard modules

(4.9)(a)  11 = X + ∑_{Y ≠X} a(Y)Y.

This is possible by general principles; there is an explicit formula due to Zuckerman ([14], 9.4.16), but we will not need this. Then

t_{∗}11 = t_{∗}(X) + ∑_{Y} a(Y)t_{∗}(Y)

= T_{n} + Z + ∑_{Y} a(Y)t_{∗}(Y). (4.9)(b)

By the results of §3 each t_{∗}(Y) is a standard module for G̃ or zero; by ([13], Proposition 2.2) T_{n} does not appear as a constituent of t_{∗}(Y) for any Y. The same holds for Z and we conclude

(4.10)(a)  a_{n} = 1 ∀n.

A similar argument applied to sgn gives

(4.10)(b)  b_{n} = 0 ∀n.

There are various ways to obtain the remaining coefficients a′_{n}, b′_{n}. In particular it follows from the character formulas in §6. In §5 we resort to a trick (see the proof of Proposition 5.4).

§5 Lifting of Unitary Representations.

We prove the main result: lifting takes an irreducible unitary representation of G to an irreducible unitary representation of G̃, up to sign, or zero. At the same time we explicitly compute the lift.

We recall the definition of some unitary representations of G.

Fix 𝔮 = 𝔩 ⊕ 𝔲, and an isomorphism ϕ : L → GL(n, ℂ) as at the end of section 4. Then (cf. for example [7], §VIII.5, example 1) for S = n(n − 1)/2 and k = 0, 1, … R_{𝔮}^{S}(χ(k)) is an irreducible representation of GL(2n, ℝ) which we denote Speh(k) (the k^{th} Speh representation). For ν ∈ iℝ let Speh(k, ν) = R_{𝔮}^{S}(χ(k, ν)) = Speh(k, 0) ⊗ χ(0, ν/2).

For k ∈ ℤ + \frac{1}{2}, L̃ has a unique (genuine) unitary character χ(k) satisfying χ(k)^{2} = χ(2k). Then for k = −\frac{1}{2}, \frac{1}{2}, \frac{3}{2}, … Speh(k) = R_{𝔮}^{S}(χ(k)) is an irreducible unitary genuine representation of G̃L(2n, ℝ). If n is even Speh(−\frac{1}{2}) = T_{n}. As before let Speh(k, ν) = Speh(k) ⊗ χ(0, ν/2). (In the literature the term Speh representation is usually reserved for the representations Speh(k) with k a positive integer.)

Let

(5.1)(a)  M = M_{1} × ⋯ × M_{r}

be a real Levi factor of G, with M_{i} = GL(m_{i}, ℝ) or M_{i} = GL(m_{i}, ℝ) × GL(m_{i}, ℝ), for each i. Fix an irreducible representation

(5.1)(b)  σ = σ_{1} × ⋯ × σ_{r}

of M.

Definition 5.2. The data (M, σ) are basic if for each 1 ≤ i ≤ r, σ_{i} is either (1) a unitary character χ(k, ν) of GL(m_{i}, ℝ),

(2) a non-unitary character χ(k, ν + t) ⊗ χ(k, ν − t) of GL(m_{i}, ℝ) × GL(m_{i}, ℝ) with 0 < t < \frac{1}{2},

(3) Speh(k, ν) of GL(m_{i}, ℝ) with k > 0,

(4) Speh(k, ν + t) ⊗ Speh(k, ν − t) of GL(m_{i}, ℝ) × GL(m_{i}, ℝ) with k > 0 and

0 < t < \frac{1}{2}. In all cases ν ∈ iℝ.

By induction in stages, (2) may be replaced by a Stein complementary series ([10], [16,§2]) of GL(2m_{i}, ℝ)

By ([16],§17) any irreducible unitary representation of G is of the form Ind_{MN}^{G}(σ) for σ a basic representation of M. (This is a reformulation of the main result of [16]; this definition of basic is not the same as ([16], Definition 3.7).)

Let π be an irreducible unitary representation of G and write

(5.3)(a)  π = Ind_{MN}^{G}(σ)

with σ a basic representation of M. By Lemma 3.3

(5.3)(b)  t_{∗}(π) = IndG̃ ; M̃N(t_{∗}^{M}(σ))

so to compute t_{∗}^{M}(π) it is enough to compute t_{∗}(σ) for σ basic, and show the induced representation is irreducible.

Proposition 5.4. Fix ν ∈ ℂ.

(1) G = GL(2n, ℝ), k = 0, 1:

t_{∗}(χ(k, ν)) = \{ \begin{matrix}Speh(−\frac{1}{2}, \frac{ν}{2} ) = T_{2n}χ(0, \frac{ν}{2} ) & k = 0 \\ Speh(\frac{1}{2}, \frac{ν}{2} ) & k = 1 \end{matrix}

(2) G = GL(2n + 1, ℝ), k = 0, 1:

t_{∗}(χ(k, ν)) = \{ \begin{matrix}T_{2n+1}(χ_{0})χ(0, \frac{ν}{2} ) & k = 0 \\ 0 & k = 1 \end{matrix}

(3) G = GL(4n, ℝ), k = 0, 2, 4 …:

t_{∗}Speh(k, ν) = (−1)^{n}IndG̃ ; MN with \{ \begin{matrix}M & = G̃L(2n, ℝ) × G̃L(2n, ℝ) \\ σ & = Speh(\frac{k+1}{2}, \frac{ν}{2} ) ⊗ Speh(\frac{k−1}{2}, \frac{ν}{2} ) \end{matrix}

(4) G = GL(4n, ℝ), k = 1, 3, 5, …:

t_{∗}Speh(k, ν) = IndG̃ ; MN(σ) with \{ \begin{matrix}M & = G̃L(2n, ℝ) × G̃L(2n, ℝ) \\ σ & = Speh(\frac{k}{2} , \frac{ν}{2} + \frac{1}{2}) ⊗ Speh(\frac{k}{2} , \frac{ν}{2} − \frac{1}{2}) \end{matrix}

(5) G = GL(4n + 2, ℝ), k = 1, 3, 5 …:

t_{∗}Speh(k, ν) = IndG̃ ; MN(σ) with \{ \begin{matrix}M & = G̃L(2n + 2, ℝ) × G̃L(2n, ℝ) \\ σ & = Speh(\frac{k}{2} , \frac{ν}{2} ) ⊗ Speh(\frac{k}{2} , \frac{ν}{2} ). \end{matrix}

(6) G = GL(4n + 2, ℝ), k = 0, 2, 4, …: t_{∗}Speh(k, ν) = 0.

Proof. By (2.7) we may take ν = 0. Statements (3-5) are reformulations of Lemmas 3.21 and 3.22, using ([16],§17). For example in (3) consider the Levi factors

L = GL(2n, ℂ)
L′ = GL(n, ℂ) × GL(n, ℂ)
M = GL(2n, ℝ) × GL(2n, ℝ)

with parabolic subalgebras and functors RG̃ ; L̃, RL̃ ; L̃′ etc. (in degree dim(𝔲 ∩ 𝔨) in each case) as in section 3. Note that RL̃ ; L̃′ and RG̃ ; M̃ are real parabolic induction. Then:

t_{∗}Speh(k) = RG̃ ; L̃RL̃ ; L̃′(χ(\frac{k+1}{2}) ⊗ χ(\frac{k−1}{2})) (Lemmas 3.21 and 3.22)

= RG̃ ; L̃′(χ(\frac{k+1}{2}) ⊗ χ(\frac{k−1}{2})) (Induction in Stages)

= RG̃ ; M̃RM̃ ; L̃′(χ(\frac{k+1}{2}) ⊗ χ(\frac{k−1}{2})) (Induction in Stages)

= RG̃ ; M̃(Speh(\frac{k+1}{2}) ⊗ Speh(\frac{k−1}{2}))

(In each case by vanishing except in one degree the induction in stages spectral sequence collapses ([7], Corollary 11.86)). Statements (4) and (5) are similar.

Statement (2) follows from (4.8) and (4.10)(b), and (6) follows from Lemma 3.21(4), or (2.6)(a).

Now consider (1): taking ν = 0 we need to compute t_{∗}(11) and t_{∗}(sgn) for GL(2n, ℝ). We claim a′_{2n} = 0, b′_{2n} = 1 in (4.8). We use the trick of inducing 11⊗sgn up to GL(4n, ℝ). Consider the identity ([16],Proposition 17.10a) on GL(4n, ℝ):

(5.5)  Speh(0) = Ind_{MN}^{G}(11 ⊗ sgn)

with M = GL(2n, ℝ) × GL(2n, ℝ). Apply t_{∗} to both sides. By Proposition 5.4(3), t_{∗} of the left hand side is:

(5.6)(a)  IndG̃ ; M̃N(Speh(\frac{1}{2}) ⊗ Speh(−\frac{1}{2})).

By Lemma 3.3 and (3.10), t_{∗} applied to the right hand side equals:

(5.6)(b)  IndG̃ ; M̃N([Speh(−\frac{1}{2}) + a′_{2n}Speh(\frac{1}{2})] ⊗ b′_{2n}Speh(\frac{1}{2})).

Setting (a) equal to (b) gives a′_{2n} = 0, b′_{2n} = 1. This completes the proof.

Proposition 1.5 is a special case of Proposition 5.4.

Theorem 5.7. For π an irreducible unitary representation of GL(n, ℝ), t_{∗}(π) is an irreducible unitary representation, up to sign, or zero. It may be computed explicitly by Proposition 5.4 and Lemma 3.3.

Proof. Write π = Ind_{MN}^{G}(σ) and t_{∗}(π) = IndG̃ ; M̃N(t_{∗}^{M}(σ)) as in (5.3); in particular σ is basic. Assume t_{∗}^{M}(σ) ≠ 0. Write t_{∗}^{M}(σ) = σ̃_{1} ⊗ ⋯ ⊗ σ̃_{r}, with each σ̃_{i} given explicitly by Proposition 5.4.

Each σ̃_{i} is unitary except for the case of

(5.8)  σ̃_{i} = Speh(\frac{k}{2} , \frac{t}{2}) ⊗ Speh(\frac{k}{2} , −\frac{t}{2}) (0 < t < \frac{1}{2})

on M̃_{i} = G̃L(m_{i}, ℝ) × G̃L(m_{i}, ℝ) (up to a unitary character of M̃_{i} which we may disregard.) In this case the induced representation

Ind_{M̃_{i}N}^{G̃L(2m_{i},ℝ)}(σ̃_{i})

is irreducible and unitary, by the main result of [3] (Theorem 5.7). (This is the situation of Lemma 3.21(2) and Proposition 5.4(4).) The Theorem follows from induction in stages and the fact that unitary induction for G̃ preserves irreducibility.

Remark 5.9. Fix n even, G = GL(2n, ℝ) and k = 0, 1. For t ∈ ℝ let π(t) be the representation of G induced from

(5.10)(a)  χ(k, t) ⊗ χ(k, −t)

on GL(n, ℝ) × GL(n, ℝ). By Proposition (5.4)(4) π̃(t) = t_{∗}(π(t)) is the representation of G̃ induced from

(5.10)(b)  Speh(±\frac{1}{2}, \frac{t}{2}) ⊗ Speh(±\frac{1}{2}, −\frac{t}{2})

on G̃L(n, ℝ) × G̃L(n, ℝ). Now π(t) is irreducible and unitary for |t| < \frac{1}{2}, reducible for |t| = \frac{1}{2}, and irreducible, non-unitary for \frac{1}{2} < |t| < 1. On the other hand π̃(t) is irreducible and unitary for |t| < 1. Therefore the irreducible unitary representation π̃(t) of G̃ is the image of a reducible representation of G if |t| = \frac{1}{2}, and an irreducible non-unitary representation if \frac{1}{2} < |t| < 1.

A similar phenomenon holds for GL(n, ℂ) [11].

Remark 5.10. The character of Speh(k) may be computed by writing the character χ(k) of GL(n, ℂ) in terms of standard modules and induction by stages, as in the proof of Theorem 3.11. We state the result without proof since we do not need it elsewhere. The case k ≥ n is in [12].

Let G = GL(2, ℝ) for the moment. Write Speh(k, ν) = R_{𝔮}^{0}(χ(k, ν)) as at the beginning of this section; this is the relative discrete series representation with infinitesimal character (\frac{k+ν}{2} , \frac{k−ν}{2} ). For k ∈ ℤ let

(5.11)(a)  π(k, ν) = R̂_{𝔮}(χ(k, ν)).

Now consider the Levi factor M = M_{1} × ⋯ × M_{n} of G = GL(2n, ℝ), with M_{i} = GL(2, ℝ) for all i. Let λ_{i} = \frac{n+1}{2} − i. For w ∈ S^{n} and 1 ≤ i ≤ n let

(5.11)(b)  σ(i, w) = π(λ_{i} − λ_{wi} + k, λ_{i} + λ_{wi}).

This is a virtual module for M_{i}, and is a relative discrete series representation if λ_{i} − λ_{wi} + k ≥ 1. Then

(5.11)(c)  Speh(k) = ∑_{w∈S^{n}} sgn(w)Ind_{MN}^{G}(σ(1, w) ⊗ ⋯ ⊗ σ(n, w)).

If λ_{i} − λ_{wi} + k ≥ 1 for all w, i.e. k ≥ n, then each σ(i, w) is a relative discrete series representation. In this case the terms on the right hand side are standard modules, and this is what is usually meant by a “character formula”. Nevertheless (5.11)(c) holds for all k ≥ 0, and a formula for the character of Speh(k) restricted to any Cartan subgroup may be derived from this. (The Speh representations form a coherent family, so these formulas are uniform in k.)

Formula (5.11)(c) holds, with the obvious changes, for the Speh representations of G̃ (k = −\frac{1}{2}, \frac{1}{2}, \frac{3}{2}, …).

§6. Character Formulas.

We compute the character of T_{n} = t_{∗}(11) and Speh(\frac{1}{2}) = t_{∗}(sgn). This is very similar to some computations in the complex case ([6], Proposition 5.9, [11]). These formulas are not needed in the rest of the paper.

Fix a θ-stable Cartan subgroup T of G and positive roots Δ^{+} of 𝔱 in 𝔤 satisfying

(6.1)  α ∈ Δ_{cx}^{+} ⇒ θα ∉ Δ^{+}

(such a choice always exists.) Let ρ ∈ 𝔱^{∗} be one-half the sum of the positive roots as usual and let

(6.2)  |e^{ρ}(g)| = |e^{2ρ}(g)|^{\frac{1}{2}} (g ∈ T)

(ρ may fail to exponentiate to T if n is even.)

Note that α/2 exponentiates to T and for g a regular element of G let

(6.3)  Γ(g) = ∏_{α∈Δ_{im}^{+}} \frac{|(1 − e^{−α/2}(g))|}{(1 − e^{−α/2}(g))}.

With Δ(g) as in §2 an elementary computation shows

(6.4)  Δ(g) = |e^{ρ}(g)|Γ(g^{2}) ∑_{w∈W} sgn(w)e^{wρ−ρ}(g)

for all g ∈ T such that e^{α}(g) < 1 for all α ∈ Δ_{re}^{+}. (This is where we use (6.1)).

Let W = W (𝔱, 𝔤) and let W (ρ/2) be the integral Weyl group for ρ/2, i.e. those w ∈ W for which wρ/2 − ρ/2 is a sum of roots. Let M be the centralizer of the split component of T in G, W_{M} = W (𝔱, 𝔪) and W′(ρ/2) the group generated by W (ρ/2) and W_{M}.

Proposition 6.5. Let g be a regular semisimple element of T̃ ∩ t(G). Assume (after conjugating by the Weyl group if necessary) e^{α}(g) < 1 for all α ∈ Δ_{re}^{+}. Then

t_{∗}(Θ_{11})(g) = c_{T} \frac{|e^{ρ/2}(g)|Γ(g)}{Δ(g)} ∑_{w∈W′(ρ/2)} sgn(w)e^{wρ/2−ρ/2}(g).

Here c_{T} = 2^{[n/2]−n+r} where r is the split rank of T.

The terms |e^{ρ/2}| = |e^{2ρ}|^{\frac{1}{4}} , Γ and e^{wρ/2−ρ/2} all exponentiate to T, so the righthand side could be written with p(g) in place of g.

Proof. Fix h_{0} satisfying t(h_{0}) = g. Then by (2.1)(b)

{h | t(h) = g} = {th_{0} | t ∈ T_{2}}

where T_{2} is the elements of T of order 2. Therefore letting c = 2^{[n/2]−n}

(6.6)  t_{∗}(θ_{11})(g) = c ∑_{t∈T_{2}} \frac{Δ(th_{0})}{Δ(g)}.

Now for t ∈ T_{2} |e^{ρ}(th_{0})| = |e^{ρ}(h)| since e^{2ρ} is a character of T. Also for all w e^{wρ−ρ}(th_{0}) = e^{wρ−ρ}(t)e^{2ρ−ρ}(h_{0}). Therefore (6.6) gives

(6.7)  t_{∗}(θ_{11})(g) = c|e^{ρ}(h_{0})|Γ(h_{0}^{2}) ∑_{w∈W} ∑_{t∈T_{2}} e^{wρ−ρ}(t)sgn(w)e^{wρ−ρ}(h_{0}).

By orthogonality of characters on the group T_{2} the inner sum is zero unless e^{wρ−ρ} is trivial on T_{2}. It is straightforward to see this condition is equivalent to w ∈ W′(ρ/2). Therefore

(6.8)  t_{∗}(θ_{11})(g) = c|e^{ρ}(h_{0})|Γ(h_{0}^{2})2^{r} ∑_{w∈W′(ρ/2)} sgn(w)e^{wρ−ρ}(h_{0}).

The proof follows upon noting that Γ(h_{0}^{2}) = Γ(g) and e^{wρ−ρ}(h_{0}) = e^{wρ/2−ρ/2}(g).

Assume n is even and replace 11 by sgn; the same computation holds, with an extra term sgn(t) on the right hand side of (6.7). Noting that e^{w_{0}ρ−ρ}(t) = e^{−2ρ}(t) = sgn(t), for w_{0} the long element of the Weyl group and t ∈ T_{2}, this gives

(6.9)  t_{∗}(Θ_{sgn})(g) = c_{T} \frac{|e^{ρ/2}(g)|Γ(g)}{Δ(g)} ∑_{w∈W′(ρ/2)} sgn(w)e^{ww_{0}ρ/2−ρ/2}(g).

References.

[1] T. Bröcker & T. tom Dieck, Representations of Compact Lie Groups, Graduate Texts in Mathematics, Springer-Verlag, Berlin-Heidelberg-New York, 1985.

[2] Y. Flicker, “Automorphic Forms on Covering Groups of GL(2),” Invent. Math. 57 (1980), 119–182.

[3] J.-S. Huang, “The unitary dual of the universal covering group of GL(n, ℝ),” Duke Math. J. 61 (1990), 705–745.

[4] D. A. Kazhdan & Y. Flicker, “Metaplectic Correspondence,” Inst. Hautes Études Sci. Publ. Math. 64 (1986), 53–110.

[5] D. A. Kazhdan & S. J. Patterson, “Metaplectic Forms,” Inst. Hautes Études Sci. Publ. Math. (1984).

[6] D. A. Kazhdan & S. J. Patterson, “Towards a Generalized Shimura Correspondence,” Adv. in Math. 60 (May 1986), 161–23.

[7] A. Knapp & D. Vogan, Cohomological Induction and Unitary Representations, Princeton University Press, Princeton, 1995.

[8] H. Matsumoto, “Sur les sous groupes arithmétiques des groupes semi–simples déployeś,” Ann. Sci. École Norm. Sup. (4) (1969).

[9] R. Ranga Rao, “On some explicit formulas in the theory of the Weil representation,” Pacific J. Math. 157 (1993), 335–371.

[10] E. M. Stein, “Analysis in matrix spaces and some new representations of SL(n, ℂ),” Ann. of Math. (2) 86 (1967), 461–490.

[11] M. Tadic, “Correspondence on Characters of Irreducible Unitary Representations of GL(n, ℂ),” preprint.

[12] M. Tadic, “On Characters of Irreducible Unitary Representations of General Linear Groups,” preprint.

[13] D. Vogan, “Irreducible Characters of Semisimple Lie Groups I,” Duke Math. J. 46 (1979), 61–108.

[14] D. Vogan, Representations of Real Reductive Lie Groups, Birkhäuser, Boston, 1981.

[15] D. Vogan, “Singular Unitary Representations,” in Non-commutative harmonic analysis and Lie groups, J. Carmona & M. Vergne, eds., Lecture notes in mathematics #880, 1981, 506–535.

[16] D. Vogan, “The Unitary Dual of GL(n) Over an Archimedean Field,” Invent. Math. 83 (1986), 449–505.

[17] D. P. Zhelobenko, Analyse harmonique sur les groupes semi-simples complexes, Nauka, Moscow, 1974.

Math Dept, University of Maryland, College Park, MD 20742

E-mail address: jda@math.umd.edu

Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

E-mail address: mahuang@uxmail.ust.hk