View source for Segal-Shale-Weil representation

A representation of groups arising in both number theory and in physics. For number theorists, the seminal paper is that of A. Weil, [[#References|[a1]]]. He cites earlier papers of I. Segal and D. Shale as precedents, and the deep work of C.L. Siegel on theta-series as inspiration.

Let $H$ be a [[Group|group]] with centre $Z$ such that $A=H/Z$ is Abelian, and let $\chi$ be a unitary character of $Z$ (cf. also [[Character of a group|Character of a group]]). If $\overline{a},\overline{b}\in A$, choose representatives $a,b\in H$ and note that $(\overline{a},\overline{b})=\chi(aba^{-1}b^{-1})$ is independent of the choice of representatives. This is a skew-symmetric bilinear pairing $A\times A\to\textbf{C}^{\times}$. One assumes that this pairing is non-degenerate. The Stone–von Neumann theorem asserts that $H$ has a unique [[Irreducible representation|irreducible representation]] $\pi$ with central character $\chi$. Furthermore, the representation may be constructed as follows. Let $L$ be a Lagrangian subgroup, that is, any subgroup of $H$ containing $Z$ such that $L/Z$ is a maximal subgroup of $A$ on which the form $(,.,)$ is trivial. Extend $\chi$ to $L$ in an arbitrary manner, then induce. This gives a model for $\pi$.

Let $G$ be a group of automorphisms of $H$ which acts trivially on $Z$ (cf. also [[Automorphism|Automorphism]]). If $g\in G$, the Stone–von Neumann theorem implies that $\Box\:\pi\cong\pi$. Let $\omega(g):\pi\to\Box$ be an intertwining mapping, well defined up to constant multiple (cf. also [[Intertwining operator|Intertwining operator]]). Then $\omega$ is a [[Projective representation|projective representation]] of $G$.

For example, let $F$ be a [[Local field|local field]] and let $W$ be a [[Vector space|vector space]] over $F$ endowed with a non-degenerate skew-symmetric bilinear form $\langle.,.\rangle$. Its dimension $2n$ is even, and the automorphism group of the form is the [[Symplectic group|symplectic group]] $\text{Sp}(2n,F)$. One can construct a "Heisenberg group" $H=W\oplus F$ with the multiplication $(w,x)(w',x')=(w+w',x+x'+\langle w,w'\rangle)$. Choosing any non-trivial additive character $\chi_0$ of $F$, let $\chi(w,x)=\chi_0(x)$. Then the hypotheses of the Stone–von Neumann theorem are satisfied. As the Lagrangian subgroup of $H$ one may take $V\oplus F$, where $V$ is any maximal isotropic subspace of $H$. Then the induced model of $\pi$ described above may be realized as the Schwartz space $S(V)$. The Segal–Shale–Weil representation is the resulting projective representation of $\text{Sp}(2n,F)$. It may be interpreted as a genuine representation of a covering group $\widetilde{\text{Sp}}(2n,F)$, the so-called metaplectic group.

Now let $F$ be a [[Global field|global field]], $A$ its adèle ring (cf. also [[Adèle|Adèle]]), and let $V$ and $W$ be as before. Then one may construct a similar representation $\omega$ of $\text{Sp}(2n,A)$ on the Schwartz space $S(A\otimes V)$. If $\Phi\in S(A\otimes V)$, let $\Lambda(\Phi)=\sum_{v\in V}\Phi(v)$. This linear form is invariant under the action of $\text{Sp}(2n,F)$, generalizing the [[Poisson summation formula|Poisson summation formula]]. This implies that the representation $\omega$ is automorphic. The corresponding automorphic forms are theta-functions (cf. [[Theta-function|Theta-function]]), having their historical origins in the work of C.G.J. Jacobi and Siegel. As Weil observed, the automorphicity of this representation is closely related to the [[Quadratic reciprocity law|quadratic reciprocity law]].

Later authors, notably R. Howe [[#References|[a2]]], have emphasized the theory of dual reductive pairs. When a pair of reductive groups $G_1\times G_2$ embeds in $\text{Sp}(2n)$, each being the centralizer of the other (cf. also [[Centralizer|Centralizer]]), then $\omega$ sets up a correspondence between representations of $G_1$ and representations of $G_2$. This works at the level of automorphic forms and gives instances of Langlands functoriality, including some historically important ones such as quadratic base change (cf. also [[Base change|Base change]]). See [[#References|[a3]]]. The use of the Weil representation in [[#References|[a4]]] to construct automorphic forms and representations may be understood as arising from the dual reductive pairs $O(2)\times\text{SL}(2)$ and $O(4)\times\text{SL}(2)$. The dual pair $O(3)\times\widetilde{\text{SL}}(2)$ underlies the important work of J.-L. Waldspurger [[#References|[a5]]] on automorphic forms of half-integral weight.

In recent years (as of 2000) it has been noted that since the Segal–Shale–Weil representation is the minimal representation of $\widetilde{\text{Sp}}(2n)$, that is, the representation with smallest Gel'fand–Kirillov dimension, minimal representations of other groups can play a similar role. Many interesting examples may be found in the exceptional groups (cf. also [[Lie algebra, exceptional|Lie algebra, exceptional]]). The possibly first paper where this phenomenon was noted was [[#References|[a6]]]. Many interesting examples come from the exceptional groups. There is much current literature on this subject, but for typical papers see [[#References|[a7]]] and [[#References|[a8]]]. Dual pairs in the exceptional groups were classified in [[#References|[a9]]].

For further references see [[#References|[a10]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Weil, "Sur certains groupes d'opérateurs unitaires" ''Acta Math.'' , '''111''' (1964) pp. 143–211 (Also: Collected Works, Vol. 3) {{MR|0165033}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R.E. Howe, "$\theta$-series and invariant theory" , ''Automorphic forms, representations and $L$-functions'' , ''Proc. Symp. Pure Math.'' , '''33:1''' , Amer. Math. Soc. (1977)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> S. Rallis, "Langlands' functoriality and the Weil representation" ''Amer. J. Math.'' , '''104''' : 3 (1982) pp. 469–515</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> H. Jacquet, R.P. Langlands, "Automorphic forms on $GL(2)$" , ''Lecture Notes in Mathematics'' , '''114''' , Springer (1970)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> J.-L. Waldspurger, "Sur les coefficients de Fourier des formes modulaires de poids demi-entier" ''J. Math. Pures Appl.'' , '''60''' (1981) pp. 375–484 {{MR|0646366}} {{ZBL|0431.10015}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> D. Kazhdan, "The minimal representation of $D_4$" , ''Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (Paris, 1989)'' , Birkhäuser (1990) pp. 125–158</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> D. Ginzburg, S. Rallis, D. Soudry, "A tower of theta correspondences for $G_2$" ''Duke Math. J.'' , '''88''' (1997) pp. 537–624</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> B.H. Gross, G. Savin, "The dual pair $\text{PGL}_3\times G_2$" ''Canad. Math. Bull.'' , '''40''' : 3 (1997) pp. 376–384</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> H. Rubenthaler, "Les paires duales dans les algèbres de Lie réductives" ''Astérisque'' , '''219''' (1994)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> D. Prasad, "A brief survey on the theta correspondence" , ''Number theory'' , ''Contemp. Math.'' , '''210''' , Amer. Math. Soc. (1998) pp. 171–193</TD></TR></table>