– The two special unipotent representations are of the form

\[ X _ { \mathcal { O } } ^ { + } = \frac { 1 } { 2 } ( R _ { e } + R _ { s } ) , ~ X _ { \mathcal { O } } ^ { - } = \frac { 1 } { 2 } ( R _ { e } - R _ { s } ) . \]

We now link the special unipotent representations obtained in the previous example with \( \theta ( \mathrm { t r i v } ) \),\( \theta ( \mathrm { d e t } ) \).

Proposition 3.3. As \( ( { \mathfrak { g } } _ { \mathbb { C } } , K _ { \mathbb { C } } ) \cdot \)-modules,

\[ X _ { \mathcal { O } } ^ { + } \cong \theta ( \mathrm { t r i v } ) , ~ X _ { \mathcal { O } } ^ { - } \cong \theta ( \mathrm { d e t } ) \]

Proof. The proposition can be proved directly by tracing along the lines of Corollary 5.24 in [5] and Theorem 2.1. We present another proof here. Note that the infinitesimal charac-ter of \( X _ { \mathcal { O } } ^ { + } \) and \( X _ { \mathcal { O } } ^ { - } \) are both \( W \times W \)-conjugacy class of) \( ( \lambda _ { \mathcal { O } } , \lambda _ { \mathcal { O } } ) \). By Proposition 1.4, they must be of the form \( L \left( \begin{array} { c } { { \dot { \lambda _ { \mathcal { O } } } } } \\ { { w \lambda _ { \mathcal { O } } } } \end{array} \right) \) for some \( w \in W \). For \( X _ { \mathcal { O } } ^ { + } \),

\[ X _ { \mathcal { O } } ^ { + } \cong \frac { 1 } { 2 } ( R _ { e } + R _ { s } ) = \frac { 1 } { 2 } [ ( X \left( \begin{array} { l } { \lambda _ { \mathcal { O } } } \\ { \lambda _ { \mathcal { O } } } \end{array} \right) + \ldots ) + ( X \left( \begin{array} { l } { \lambda _ { \mathcal { O } } } \\ { \lambda _ { \mathcal { O } } } \end{array} \right) + \ldots ) ] = X \left( \begin{array} { l } { \lambda _ { \mathcal { O } } } \\ { \lambda _ { \mathcal { O } } } \end{array} \right) + \ldots , \]

where the remaining terms are of the form \( X \left( \begin{array} { c } { { \lambda \mathcal { O } } } \\ { { w \lambda \mathcal { O } } } \end{array} \right) \! . \), \( l ( w ) > \) 0. Hence its lowest \( K _ { \mathbb { C } } \)-type is \( \lambda _ { \mathcal { O } } - \lambda _ { \mathcal { O } } = ( 0 ^ { n } ) \). On the other hand, the only \( L \left( \begin{array} { c } { { \lambda \mathcal { O } } } \\ { { w \lambda \mathcal { O } } } \end{array} \right) \) having lowest \( K _ { \mathbb { C } } \)-type \( ( 0 ^ { n } ) \) is \( L \left( \begin{array} { c } { { \lambda _ { \mathcal { O } } } } \\ { { \lambda _ { \mathcal { O } } } } \end{array} \right) . \) By (Remark (b) after) Proposition 1.4 and Theorem 2.1,

\[ \begin{array} { r } { X _ { \mathcal { O } } ^ { + } \cong L \left( \begin{array} { l } { \lambda _ { \mathcal { O } } } \\ { \lambda _ { \mathcal { O } } } \end{array} \right) \cong L \left( \begin{array} { l l l } { p + q , \dots , 2 , 1 ; } & { p - 1 , \dots , 1 , 0 } \\ { p + q , \dots , 2 , 1 ; } & { p - 1 , \dots , 1 , 0 } \end{array} \right) \cong \theta ( \mathrm { t r i v } ) . } \end{array} \]

For \( X _ { \mathcal { O } } ^ { - } \), a direct computation shows that the term

\[ X \left( \begin{array} { c } { { \lambda _ { \mathcal { O } } } } \\ { { w _ { p } \lambda _ { \mathcal { O } } } } \end{array} \right) : = X \left( \begin{array} { c c c c c c } { { p + q , \ldots , p + 1 ; } } & { { p , p - 1 } } & { { p - 1 , p - 2 } } & { { \ldots , } } & { { 2 , 1 , } } & { { 1 , 0 } } \\ { { p + q , \ldots , p + 1 ; } } & { { p - 1 , p } } & { { p - 2 , p - 1 } } & { { \ldots , } } & { { 1 , 2 , } } & { { 0 , 1 } } \end{array} \right) \, . \]

appears in the expression \( \begin{array} { r } { X _ { \mathcal { O } } ^ { - } \cong \frac { 1 } { 2 } ( R _ { e } - R _ { s } ) } \end{array} \) with coefficient 1. Hence \( X _ { \mathcal { O } } ^ { - } \) must have lowest \( K _ { \mathbb { C } } \)-type smaller than or equal to \( ( 1 ^ { 2 p } 0 ^ { q } ) \sim ( \lambda _ { \mathcal { O } } \! - \! w _ { p } \lambda _ { \mathcal { O } } ) \), that is, the lowest \( K _ { \mathbb { C } } \)-type must be of the form \( ( 1 ^ { 2 i } 0 ^ { n - 2 i } ) \) with \( i \leq p \). However, another direct computation shows that all terms of the form

\[ \{ X \left( \begin{array} { c } { { \lambda \mathcal O } } \\ { { w \lambda \mathcal O } } \end{array} \right) | w \in W , \; ( \lambda \mathcal O - w \lambda \mathcal O ) \sim ( 1 ^ { 2 i } 0 ^ { n - 2 i } ) , \; i < p \} \]

do not appear in the expression of \( X _ { \mathcal { O } } ^ { - } \). Therefore, the lowest \( K _ { \mathbb { C } } \)-type of \( X _ { \mathcal { O } } ^ { - } \) must be \( \left( 1 ^ { 2 p } 0 ^ { q } \right) \).

Suppose \( X _ { \mathcal { O } } ^ { - } \cong L \left( \begin{array} { c } { \lambda \mathcal { O } } \\ { w \lambda \mathcal { O } } \end{array} \right) \) for some \( w \in W \), then we have