April 3, 2002

UNITARY REPRESENTATIONS WITH HALF-INTEGER PARAMETER

1. Parametrization via Ǒ

1.1. There is an exact sequence

1 ⟶ κ ⟶ A(^{L}ϕ) ⟶ \overline{A(^{L}ϕ)} ⟶ 1.  (1.1.1)

We attach a unipotent infinitesimal character to each conjugacy class in \overline{A(^{L}ϕ)}.

1.2. Type B. The dual group is type C. A nilpotent orbit Ǒ is given by a partition

0 ≤ 2n_{0} < 2n_{1} ≤ ⋯ < 2n_{2i−1} ≤ 2n_{2i} < ⋯ ≤ 2n_{2p}.  (1.2.1)

The centralizer is G(^{L}ϕ) is a product of O(2)’s, one for each pair n_{2i−1} = n_{2i} and O(1)’s one for each remaining n_{i} > 0.

An element s ∈ G(^{L}ϕ) which gives rise to a halfinteger parameter can be written as

s = (ϵ_{1}, …, ϵ_{p}), ϵ_{i} = ±1 (1.2.2)

with ϵ_{0} = 1 if n_{0} = 0. An element s is in κ if and only if

s ∈ κ ↔ ϵ_{2i−1}ϵ_{2i} = 1.  (1.2.3)

The infinitesimal character of the unipotent representation is as follows (1) For each ϵ_{i} = 1, add (n_{i} − 1/2, …, 1/2).

(2) For ϵ_{i} = −1 order the n_{i} in increasing order and name them m_{0}, …, m_{2q}. Then add

(m_{j}, …, 1)  if j is even,

(1.2.4)

(m_{j} − 1, …, 1, 0) if j is odd.

Every coset has a representative of the form

(η_{0}, η_{0}, …, ηp − 1, η_{p−1}, 1)  if n_{0} ≠ 0,

(1.2.5)

(1, η_{0}, η_{1}, η_{1}, …, η_{p−1}, η_{p−1}, 1) if n_{0} = 0.

The infinitesimal character corresponding to this representative is called unipotent.

1.3. Type C. The dual group is type B. A nilpotent orbit Ǒ is given by a partition

0 < 2n_{0}+1 ≤ 2n_{1}+1 < ⋯ < 2n_{2i}+1 ≤ 2n_{2i+1}+1 < ⋯ < 2n_{2p}+1. (1.3.1)

The centralizer is G(^{L}ϕ) is a product of O(2)’s, one for each pair n_{2i} = n_{2i+1} and O(1)’s one for each remaining n_{i} > 0.

An element s ∈ G(^{L}ϕ) which gives rise to a halfinteger parameter can be written as

s = (ϵ_{0}, …, ϵ_{2p}),  ϵ_{i} = ±1,  ϵ_{0} · · · · · ϵ_{2p} = 1.  (1.3.2)

An element s is in κ if and only if

s ∈ κ ↔ ϵ_{2i}ϵ_{2i+1} = 1.  (1.3.3)

The infinitesimal character of the unipotent representation is as follows

(1) For ϵ_{i} = 1, order the n_{i} in increasing order and name them m_{0}, …, m_{2r}. Then add

(m_{j}, …, 1) if j is even,

(1.3.4)

(m_{j}, …, 1, 0) if j is odd.

(2) For ϵ_{i} = −1, order the n_{i} in increasing order and name them m_{0}, …, m_{2q+1}. Then add

(m_{j} + 1/2, …, 1/2) if j is odd,

(1.3.5)

(m_{j} − 1/2, …, 1/2) if j is even.

Every coset has a representative of the form

(1, η_{1}, η_{1}, …, η_{p}, η_{p}).  (1.3.6)

The infinitesimal character corresponding to this representative is called unipotent.

1.4. Type D. The dual group is type D. A nilpotent orbit Ǒ is given by a partition

0 < 2n_{0} + 1 ≤ 2n_{1} + 1 < ⋯ < 2n_{2i} + 1 ≤ 2n_{2i+1} + 1 < ⋯ < 2n_{2p+1} + 1.

(1.4.1)

The centralizer is G(^{L}ϕ) is a product of O(2)’s, one for each pair n_{2i} = n_{2i+1} and O(1)’s one for each remaining n_{i} > 0.

An element s ∈ G(^{L}ϕ) which gives rise to a halfinteger parameter can be written as

s = (ϵ_{1}, …, ϵ_{2p+1}), ϵ_{i} = ±1. (1.4.2)

An element s is in κ if and only if

s ∈ κ ↔ ϵ_{2i}ϵ_{2i+1} = 1.  (1.4.3)

The infinitesimal character of the unipotent representation is as follows

(1) For ϵ_{i} = 1, order the n_{i} in increasing order and name them m_{0}, …, m_{2r+1}. Then add

(m_{j}, …, 1) if j is even,

(1.4.4)

(m_{j}, …, 1, 0) if j is odd.

(2) For ϵ_{i} = −1, order the n_{i} in increasing order and name them m_{0}, …, m_{2q+1}. Then add

(m_{j} + 1/2, …, 1/2) if j is odd,

(1.4.5)

(m_{j} − 1/2, …, 1/2) if j is even.

Every coset has a representative of the form

(1, η_{1}, η_{1}, …, η_{p}, η_{p}, 1).  (1.4.6)

The infinitesimal character corresponding to this representative is called unipotent.

2. Parametrization via O

We rely on [BV2] and [B1]. First, we prescribe the infinitesimal character λ_{O}. The main property will be that the unipotent representations (irreducible (𝔤, K) modules whose annihilator in the universal enveloping algebra is maximal with the given infinitesimal character) are unitary and in 1-1 correspondence with the irreducible characters of the component group A(O). The notation is as in [B1]. An orbit is called cuspidal if it is not induced from any proper Levi component. For special orbits whose dual is even the infinitesimal character is one half the semisimple element of the Lie triple corressponding to the dual orbit. For the other orbits we need the case-by-case analysis.

Type A. A nilpotent orbit is determined by its Jordan canonical form. It is given by a partition i.e. a sequence of numbers in decreasing order (n_{1}, …, n_{k}) that add up to n. Let (m_{1}, …, m_{l}) be the dual partition. Then the infinitesimal character is

(\frac{m_{1} − 1}{2}, …, −\frac{m_{1} − 1}{2}, …, \frac{m_{l} − 1}{2}, …, −\frac{m_{l} − 1}{2})

The orbit is induced from the trivial orbit on the Levi component GL(m_{1})× ⋯ × GL(m_{l}). The corresponding unipotent representation is spherical and induced irreducible from the trivial representation on the same Levi component. All orbits are stably trivial. Type B. A nilpotent orbit is determined by its Jordan canonical form (in the standard representation). It is parametrized by a partition (n_{1}, …, n_{k}) of 2n + 1 such that every even part occurs an even number of times. Let (m′_{0}, …, m′_{2p′}) be the dual partition (add an m′_{2p′} = 0 if necessary, in order to have an odd number of terms). If there are any m′_{2j} = m′_{2j+1} then pair them together and remove them from the partition. Then relabel the remaining columns and pair them up, the rest of the columns (m_{0})(m_{1}, m_{2}) … (m_{2p−1}m_{2p}). The members of each pair have the same parity and m_{0} is odd. Then form a parameter

\begin{matrix}(m_{0}) & ↔ (\frac{m_{0} − 2}{2}, …, 1/2), \\ (m′_{2j} = m′_{2j+1}) & ↔ (\frac{m_{2j} − 1}{2}, …, −\frac{m_{2j} − 1}{2}) \\ (m_{2i−1}m_{2i}) & ↔ (\frac{m_{2i−1}}{2} , …, −\frac{m_{2i} − 2}{2}) \end{matrix}

In case m′_{2j} = m′_{2j+1}, the nilpotent orbit is induced from a parabolic subalgebra 𝔭 with Levi component so(∗) × gl(m′_{2j}) with the trivial nilpotent on the gl factor. The component groups in G and P are equal. The unipotent representations are unitarily induced irreducible from similar parameters on the Levi component. Similarly if some m_{2i−1} = m_{2i}, then the nilpotent is induced irreducible from a nilpotent on an so(∗)×gl(\frac{m_{2i−1}+m_{2i}}{2} ) with the trivial nilpotent on the gl factor. The component groups of the centralizers in G and P coincide. The unipotent representations are again induced irreducible from the nilpotent orbit on so(∗) with partition the one for O but with m_{2i−1}, m_{2i} removed. The stably trivial orbits are the ones such that every odd sized part appears an even number of times except for the largest size. An orbit is trinagular if it has partition (1, 1, 3, 3, …, 2m−1, 2m−1, 2m+1). It is induced from the trivial nilpotent orbit on 𝔪 = gl(2) × gl(4) … gl(2m). The component group A_{P} is trivial. Type C. A nilpotent orbit is determined by its Jordan canonical form (in the standard representation). It is parametrized by a partition (n_{1}, …, n_{k}) of 2n such that every odd part occurs an even number of times. Let (m′_{0}, …, m′_{2p′}) be the dual partition (add a m′_{2p′} = 0 if necessary in order to have an odd number of terms). If there are any m′_{2j−1} = m′_{2j} pair them up and remove them from the partition. Then relabel and pair up the remaining columns (m_{0}m_{1}) … (m_{2p−2}m_{2p−1})(m_{2p}). The members of each pair have the same parity. The last one, m_{2p}, is always even. Then form a parameter

\begin{matrix}(m′_{2j−1} = m′_{2j}) & ↔ (\frac{m_{2j} − 1}{2}, …, −\frac{m_{2j} − 1}{2}) \\ (m_{2i}m_{2i+1}) & ↔ (\frac{m_{2i}}{2} , …, −\frac{m_{2i+1} − 2}{2}), \\ m_{2p} & ↔ (\frac{m_{2p}}{2} , …, 1). \end{matrix}

The nilpotent orbits and the unipotent representations have the same properties with respect to these pairs as the corresponding ones in type B. The stably trivial orbits are the ones such that every even sized part appears an even number of times. An orbit is called triangular if it corresponds to the partition (2, 2, 4, 4, …, 2m, 2m). It is induced from the trivial orbit on 𝔪 = sp(2m) × gl(1) × ⋯ × gl(2m − 1). The component group A_{P} is trivial. Type D. A nilpotent orbit is determined by its Jordan canonical form (in the standard representation). It is parametrized by a partition (n_{1}, …, n_{k}) of 2n such that every even part occurs an even number of times. Let (m′_{0}, …, m′_{2p′−1}) be the dual partition (add a m′_{2p−1} = 0 if necessary). If there are any m′_{2j} = m′_{2j+1} pair them up and remove from the partition. Then pair up the remaining columns (m_{0}m_{2p})(m_{1}, m_{2}) … (m_{2p−2}m_{2p−1}). The members of each pair have the same parity and m_{0}, m_{2p−1} are even. Then form a parameter

\begin{matrix}(m′_{2j} = m′_{2j+1}) & ↔ (\frac{m′_{2j} − 1}{2} …, −\frac{m′_{2j} − 1}{2}) \\ (m_{0}m_{2p−1}) & ↔ (\frac{m_{0} − 2}{2}, …, −\frac{m_{2p−1}}{2} ), \\ (m_{2i−1}m_{2i}) & ↔ (\frac{m_{2i−1}}{2} …, −\frac{m_{2i} − 2}{2}) \end{matrix}

The nilpotent orbits and the unipotent representations have the same properties with respect to these pairs as the corresponding ones in type B. An exception occurs when the partition is formed of pairs (m′_{2j} = m′_{2j+1}) only. In this case there are two nilpotent orbits corresponding to the partition. There are also two nonconjugate Levi components of the form gl(m′_{0}) × gl(m′_{2}) × … gl(m′_{2p′−2}) of parabolic subalgebras. There are two unipotent representations each induced irreducible from the trivial representation on the corresponding Levi component. The stably trivial orbits are the ones such that every even sized part appears an even number of times. A nilpotent orbit is triangular if it corresponds to the partition (1, 1, 3, 3, …, 2m − 1, 2m − 1). It is induced from the trivial orbit in the Levi component 𝔪 = gl(2) × ⋯ × gl(2m − 2). The component group A_{P} is trivial.

References

[AB] J. Adams, D. Barbasch The Reductive Dual Pairs Correspondence for Complex Groups J. of Func. An. vol 132, 1995

[B1] D. Barbasch The unitary dual for complex classical Lie groups Invent. Math., vol 96, 1989, pp 103-176

[B2] Unipotent representations for real reductive groups Proceedings of ICM 1990, Springer Verlag, Tokyo, 1991, pp. 769-777

[B3] Orbital integrals of nilpotent orbits volume in honor of Harish-Chandra.

[Br] Line bundles on the cotangent bundle of the flag variety Invent. Math., vol 113 Fasc. 1, 1993, pp. 1-20

[BV] N. Berline, M. Vergne Fourier transforms of orbits of the coadjoint representation Representation theory of reductive groups, Birkhäuser-Boston, Progress in mathematics vol 40, 1983, pp. 53-69

[BV1] D. Barbasch, D. Vogan The local structure of characters J. Funct. Analysis, vol. 34 no. 1, 1980, pp. 27-55

[BV2] D. Barbasch, D. Vogan Unipotent representations of complex semisimple Lie groups Ann. of Math., 1985, vol 121, pp. 41-110

[BV3] D. Barbasch, D. Vogan Weyl group representations and nilpotent orbits Representation theory of reductive groups, Birkhäuser-Boston, Progress in mathematics vol 40, 1983, pp. 21-35

[G] W. Graham, Functions on the universal cover of the principal nilpotent orbit, Inv. Math. 108 (1992), 15-27.

[HC] Harish-Chandra Fourier transform on a semisimple Lie algebra II Amer. J. of Math., vol 79, 1957, pp. 193-257

[HC1] Harish-Chandra Harmonic analysis on semisimple Lie groups Bulletin AMS. vol 76, 1970, pp. 529-551

[KP] H-P. Kraft, C. Procesi On the geometry of conjugacy classes in classical groups Comm. Math. helv., 57, 1982, pp. 539-601

[KV] A. Knapp, D. Vogan Cohomological induction and unitary representations Princeton University Press, Princeton NJ, 1995

[KLT] Frobenius splitting of cotangent bundles of flag varieties Invent. Math., 136, 1999, pp. 603-621

[LS] G. Lusztig, N. Spaltenstein Induced unipotent classes J. London Math. Soc., vol. 19, 1979, pp. 41-52

[McG] M. McGovern Rings of regular functions on nilpotent orbits and their covers Invent. Math., 97, 1989, pp. 209-217

[R] R. Rao Orbital integrals in reductive groups Annals of Math., vol. 96 no. 3, 1972, pp. 505-510

[Ro] W. Rossmann Nilpotent orbital integrals in a real semisimple Lie algebra and representations of Weyl groups Operator algebras, unitary representations, enveloping algebras and invariant theory, Progress in Mathematics 92, Birkhäuser Boston, 1990, pp. 263-287

[SV1] W. Schmid, K. Vilonen Two geometric character formulas for reductive Lie groups J. Amer. math. Soc., vol. 11 no. 4, 1998, pp. 799-867

[SV2] W. Schmid, K. Vilonen The B-V conjectures preprint

[V] D. Vogan Associated varieties and unipotent representations Harmonic Analysis on reductive groups, Progress in Mathematics vol. 101, Birkhäuser, Boston-Basel-Berlin, 1991, pp. 315-388

[W] N. Wallach Invariant differential operators on a reductive Lie algebra and Weyl group representations J. Amer. Math. Soc., vol. 6 no. 4, 1993, pp. 779-816