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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s1301801.png" /> be a [[Group|group]] with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s1301802.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s1301803.png" /> is Abelian, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s1301804.png" /> be a unitary character of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s1301805.png" /> (cf. also [[Character of a group|Character of a group]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s1301806.png" />, choose representatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s1301807.png" /> and note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s1301808.png" /> is independent of the choice of representatives. This is a skew-symmetric bilinear pairing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s1301809.png" />. One assumes that this pairing is non-degenerate. The Stone–von Neumann theorem asserts that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018010.png" /> has a unique [[Irreducible representation|irreducible representation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018011.png" /> with central character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018012.png" />. Furthermore, the representation may be constructed as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018013.png" /> be a Lagrangian subgroup, that is, any subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018014.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018015.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018016.png" /> is a maximal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018017.png" /> on which the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018018.png" /> is trivial. Extend <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018019.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018020.png" /> in an arbitrary manner, then induce. This gives a model for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018021.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018022.png" /> be a group of automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018023.png" /> which acts trivially on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018024.png" /> (cf. also [[Automorphism|Automorphism]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018025.png" />, the Stone–von Neumann theorem implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018026.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018027.png" /> be an intertwining mapping, well defined up to constant multiple (cf. also [[Intertwining operator|Intertwining operator]]). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018028.png" /> is a [[Projective representation|projective representation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018029.png" />.

For example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018030.png" /> be a [[Local field|local field]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018031.png" /> be a [[Vector space|vector space]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018032.png" /> endowed with a non-degenerate skew-symmetric bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018033.png" />. Its dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018034.png" /> is even, and the automorphism group of the form is the [[Symplectic group|symplectic group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018035.png" />. One can construct a "Heisenberg group" <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018036.png" /> with the multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018037.png" />. Choosing any non-trivial additive character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018038.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018039.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018040.png" />. Then the hypotheses of the Stone–von Neumann theorem are satisfied. As the Lagrangian subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018041.png" /> one may take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018042.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018043.png" /> is any maximal isotropic subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018044.png" />. Then the induced model of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018045.png" /> described above may be realized as the Schwartz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018046.png" />. The Segal–Shale–Weil representation is the resulting projective representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018047.png" />. It may be interpreted as a genuine representation of a covering group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018048.png" />, the so-called metaplectic group.

Now let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018049.png" /> be a [[Global field|global field]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018050.png" /> its adèle ring (cf. also [[Adèle|Adèle]]), and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018052.png" /> be as before. Then one may construct a similar representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018053.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018054.png" /> on the Schwartz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018055.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018056.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018057.png" />. This linear form is invariant under the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018058.png" />, generalizing the [[Poisson summation formula|Poisson summation formula]]. This implies that the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018059.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018060.png" /> embeds in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018061.png" />, each being the centralizer of the other (cf. also [[Centralizer|Centralizer]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018062.png" /> sets up a correspondence between representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018063.png" /> and representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018064.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018066.png" />. The dual pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018067.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018068.png" />, 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, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018069.png" />-series and invariant theory" , ''Automorphic forms, representations and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018070.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018071.png" />" , ''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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018072.png" />" , ''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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018073.png" />" ''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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130180/s13018074.png" />" ''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>