Using [5, Proposition 6.6], if \( \sigma _ { x } \) can be obtained as a truncated induction (see [9, Chapter 11])from the sign representation of a Weyl subgroup \( W ^ { \prime } \leq W \), i.e. \( \sigma _ { x } = j _ { W ^ { \prime } } ^ { W } ( \mathrm { s g n } ) \), then we can reduce this formula into

\[ R _ { x } : = \sum _ { w ^ { \prime } \in W ^ { \prime } } ( - 1 ) ^ { l ( w ^ { \prime } ) } X \left( \begin{array} { c } { { \lambda _ { \mathcal { O } } } } \\ { { w ^ { \prime } \lambda _ { \mathcal { O } } } } \end{array} \right) . \]

Step (4) Consequently, every unipotent representation corresponding to \( \mathcal { O } \) is parametrized by \( \pi \in \overline { { A } } ( \mathcal { O } ) ^ { \wedge } \), and has the character formula

\[ X _ { \pi } : = \frac { 1 } { | \overline { { A } } ( \mathcal { O } ) | } \sum _ { x \in \overline { { A } } ( \mathcal { O } ) } t r ( \pi ( x ) ) R _ { x } . \]

Example 3.2. We now study the special unipotent representations attached to the orbit \( \mathcal { O } = ( 2 ^ { 2 p } 1 ^ { 2 q } ) \):

– The Lusztig-Spaltenstein dual is given by \( { \cal L } { \cal O } = ( 2 p + 2 q + 1 , 2 p - 1 , 1 ) \). Hence

\[ \lambda \mathcal { O } = \frac { 1 } { 2 } \, ^ { L } h = ( p + q , p + q - 1 , \dots , p , ( p - 1 ) ^ { 2 } , \dots , 2 ^ { 2 } , 1 ^ { 2 } , 0 ) . \]

– In [20], Lusztig defined an injection

\[ \gamma ( \mathcal { O } ) : = \{ \mathcal { O } ^ { \prime } \subseteq \overline { { \mathcal { O } } } | \mathcal { O } ^ { \prime } \not \subseteq \overline { { \mathcal { O } } } _ { s p e c } \mathrm { ~ f o r ~ a n y ~ o t h e r ~ s p e c i a l ~ o r b i t ~ } \mathcal { O } _ { s p e c } \subseteq \mathcal { O } \} \hookrightarrow \overline { { A } } ( \mathcal { O } ) . \]

Consider the following composition of maps:

\[ \mathcal { O } ^ { \prime } \in \gamma ( \mathcal { O } ) \hookrightarrow x ( \mathcal { O } ^ { \prime } ) \in \overline { { A } } ( \mathcal { O } ) \overset { S t e p ( 2 ) } { \longmapsto } \sigma _ { x ( \mathcal { O } ^ { \prime } ) } \in \hat { W } . \]

Then \( \sigma _ { x ( \mathcal { O } ^ { \prime } ) } = s p ( \mathcal { O } ^ { \prime } ) \), the Springer representation of \( \mathcal { O } ^ { \prime } \).

For \( \mathcal { O } = ( 2 ^ { 2 p } 1 ^ { 2 q } ) \), \( \gamma ( O ) = \{ O , { \mathcal { O } } ^ { \prime } \} \), where \( \mathcal { O } ^ { \prime } = ( 2 ^ { 2 p - 1 } 1 ^ { 2 q + 2 } ) \), and \( A ( \mathcal { O } ) = \overline { { A } } ( \mathcal { O } ) = \mathbb { Z } / 2 \mathbb { Z } = \)\( \{ e , s \} \) with e being the identity element. So the injection above is indeed a bijection, and

\[ \mathcal { O } \leftrightarrow e \; ; \; \; \mathcal { O } ^ { \prime } \leftrightarrow s . \]

According to the algorithm of computing Springer representations given in Section 7 of[24],

\[ \sigma _ { e } = ( 1 ^ { p } , 1 ^ { p + q } ) = j _ { D _ { p } \times C _ { p + q } } ^ { C _ { n } } ( \mathrm { s g n } ) , \; \; \sigma _ { s } = ( 1 ^ { p + q + 1 } , 1 ^ { p - 1 } ) = j _ { D _ { p + q + 1 } \times C _ { p - 1 } } ^ { C _ { n } } ( \mathrm { s g n } ) . \]

– The two reduced formula is of the form

\[ ( 3 ) \qquad \quad R _ { e } = \sum _ { w \in W ( D _ { p } \times C _ { p + q } ) } ( - 1 ) ^ { l ( w ) } X \left( \begin{array} { c c c } { { p - 1 , \ldots , 1 , 0 ; } } & { { p + q , \ldots , 2 , 1 } } \\ { { w ( } } & { { p - 1 , \ldots , 1 , 0 ; } } & { { p + q , \ldots , 2 , 1 ) } } \end{array} \right) , \]

\[ ( 4 ) \qquad R _ { s } = \sum _ { w ^ { \prime } \in W ( D _ { p + q + 1 } \times C _ { p - 1 } ) } ( - 1 ) ^ { l ( w ^ { \prime } ) } X \left( \begin{array} { c } { { p + q , \dots , 1 , 0 ; } } \\ { { w ^ { \prime } ( } & { { p + q , \dots , 1 , 0 ; } } \end{array} \begin{array} { c } { { p - 1 , \dots , 2 , 1 } } \\ { { p - 1 , \dots , 2 , 1 } } \end{array} \right) . \]

(Here we have used Remark (a) after Proposition 1.4.)

– 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