Minimal representations for simply-laced groups
All simply-laced groups G (and some non-simply laced ones) admit a
representation in a quantum phase space of minimal dimension given
by the dimension of the smallest nilpotent orbit in G. This
representation generalizes the Schrodinger (or metaplectic) representation
of the symplectic group Sp(n,R) in the phase space of the n-dimensional
harmonic oscillator.
The paper ``Minimal representations, spherical vectors, and exceptional
theta series I'' (D. Kazhdan, B. Pioline and A. Waldron,
hep-th/0107222) presents an explicit construction of this
representation for all ADE groups and its spherical vector, i.e., the
wave function annihilated by all compact generators.
This result is an important ingredient in constructing theta series, and
conjecturally
encodes information about the quantum BPS membrane.
From this site you can download the list of generators for all groups,
either in Form output format, or
in Mathematica input format:
Form:
[ A2 |
A3 |
A5 |
D4 |
D5 |
E6 |
E7 |
E8 ]
Mathematica:
[ A2 |
A3 |
A5 |
D4 |
D5 |
E6 |
E7 |
E8 ]
You may also download all at once in a gzipped
tar file.
The positive generators are denoted a#, b#, c#,om for the positive
roots,
where a# correspond to the grade-0 roots, b# the grade one
roots represented as momenta, c# the grade one roots represented as
positions, and om the grade 2 highest root. The notation am#,bm#,cm#,mom
is used for the negative ones. H# denote the Cartan generators wrt the
simple roots, and Hom the Cartan generator for the highest root.
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