Let and let be the nilpotent radical of the standard Borel subgroup, i.e., is the subgroup of consisting of upper triangular matrices with 1’s on the diagonal. Given a non-trivial character , we define a character of by

This defines a one-dimensional representation of . One can show that the induced representation is multiplicity-free, i.e., when we decompose into a sum of irreducible constituents,

we have for all . Each constituent is called a *generic* irreducible representation. There are many reasons for calling them *generic*. One is that no matter which non-trivial character we choose, we obtain the same induced representation up to equivalence. One can see this by looking at the induced character values on each conjugacy class. These values will be polynomials in and, once simplified, do not depend on the character . Moreover, this *Gelfand-Graev representation* contains *most* irreducible representations of the group.

To see this, we can compare the number of irreducible constituents of with the number of irreducible representations of . By a standard result in the representation theory of finite groups, the number of irreducible representations of is equal to its number of conjugacy classes.

J. A. Green wrote a paper about the irreducible characters of in which he gives a generating function for the number of classes of :

Let be the number of partitions of *n*. We define the partition function

The generating function for the number of classes of is

where

if the number of irreducible polynomials of degree over .

Comparing with the character inner product , one can see that most irreducible representations of are generic.

**General symplectic groups**

We have a similar story for . Again, let be nilpotent radical of the standard Borel subgroup, i.e., is the subgroup of consisting of upper triangular matrices with 1’s on the diagonal. Given a non-trivial character , we define a character of by

and induce to . The induced representation is multiplicity-free and contains most of the irreducible representations of . The case was made explicit in my thesis.

**Conjugacy classes of **

The conjugacy classes can be found by using a paper by Wall to compute the classes of , where denotes the center of . The list of conjugacy classes of is then used to determine the classes of . Let’s investigate how the class representatives of lead to representatives for the classes of .

Consider the natural projection map from to given by

Let . If , then, by taking multipliers on each side, it is clear that the multiplier of is equal to the multiplier of . Moreover, under the projection map, we have

So if two elements are conjugate in , they must be conjugate in . The list of class representatives in , when pulled back to , hit class representatives of all the conjugacy classes of . Suppose now that two elements are conjugate in , i.e. , for some . Then, for some ,

Taking multipliers on both sides of the equation above, we have . So if the multiplier of is a square, then the multiplier of is a square and if the multiplier of is a non-square, then the multiplier of is a non-square. Write and in the following way

with , , and . So . If and are conjugate, then . Then , or . So , i.e., the multipliers can only differ by a minus sign.

It is possible that an element of is conjugate to . An example is

The centralizers are somewhat affected when we pull back our representatives in to . There are two types of pullbacks. The first type consists of elements such that for any . The second type consists of elements such that for some . Let denote the centralizer of in and let denote the centralizer of in .

*Type 1*

Let be of the first type, i.e., is not conjugate to . Let . Then . When pulled back to , , with . since is not conjugate to . So , , and . We get a short exact sequence

Therefore .

*Type 2*

Let be of the second type, i.e., is conjugate to . Define the set . Fix . is not a group, but there is a bijection of sets , given by the map . Given , either or . maps onto via the projection map. Moreover, is a group with respect to matrix multiplication and the projection map is a group homomorphism. is a subgroup of of index 2. We get a short exact sequence

Then . Also, . So .

Thus, given a class representative , we pull it back to . Then, we determine if is of Type 1 or Type 2. If it is of Type 1, then there are conjugacy classes , for , each of order . If the pullback is of Type 2, then there are conjugacy classes , with

Each of these classes is of order

Some of the conjugacy classes of are given in the following table. Let .

Notation | Class representative | Number of classes | Order of centralizer | |

, |
||||

, |
||||

, |
||||

, |
||||

, |

We find that there are conjugacy classes.

**The conjugacy classes of **

Every element can be written uniquely in the form

with . The order of is therefore . The multiplier of the matrix given above is . The conjugacy classes of are listed in the following table.

Notation | Class representative | Number of classes | Order of centralizer | |

, |
||||

, |
||||

, for some |
||||

, |
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, |
||||

, for some |
||||

, |

**Generic representations of **

The character table of is

Conjugacy class | character value | |

We compute that there are irreducible generic representation, which is the majority of the total irreducible representations.

**References**

J. Breeding II, *Irreducible non-cuspidal characters of *, Ph.D. thesis, University of Oklahoma, Norman, OK, 2011.

D. Bump, *Lie Groups*, Springer, 2004.

W. Fulton and J. Harris, *Representation Theory, A First Course*, Springer, 2004.

J.A. Green, *The Characters of the Finite General Linear Groups*, Transactions of the American Mathematical Society, **80**, 2 (1955), 402–447.

G.E. Wall, *On the Conjugacy Classes in the Unitary, Symplectic and Orthogonal Groups*, Journal of the Australian Mathematical Society, **3** (1963), 1–62.

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