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{\bold $SL_{2}({\Bbb R})$}

\q
Structure
$G=PK$, $G=KAK$, ${\cal H}$,
$P\\G$, ${\goth g}$ etc., $A$, $N$, $K$, $\theta$, $PwP \cup P$.
Invariance of $dx^2 + dy^2/y^2$,
$dx xy / y^2$, $y^2(d/dx^2 + d/dy^2) = \Delta$.
Casimir polynomials make up $Z({\goth g})$.

\q
Example; $V = C^{\infty}(P\\G)$.  Restriction to $K$.  Dual.
How ${\goth g}$ acts (to be done later for general PS).
$f \mapsto f(1)$ and Taylor series.  
$\Delta = 0$ on ${\cal H}$-realization,  
how Casimir acts (for general p.s.).  Boundary values.

\q
Definition of admissible.  f.g. over $U({\goth n})$
and f.g. over $U({\goth g})$.
Skip realizability for now.

\q
Frobenius reciprocity.

\q
Matrix coefficients and realizability.
Differential equations.

\q
Asymptotics and $V/{\goth n}V$.
F.d. spaces

\q
Bruhat filtration and i.o.s

\q
Explicit formula for asymptotics of p.s.

\q
Plancherel measure for wave functions.

\q
Paley-Wiener theorems.

\q
Endoscopy, characters, orbital integrals.

\q
Canonical $G$-models, smooth and analytic.

\q
Beilinson-Bernstein, $D$-modules, $V/{\goth n}V$.

\q
Whittaker models.

\q
WF sets?

\q
Asymptotics and details of $V_{[{\goth n}]}$.

\q
Milicic' notes on Plancherel measure?

\q
Langlands' notes on $SL_{2}$.

\q
Weil reps.

\q 
Lusztig-Vogan, KL  conjecture.

\q
Hodge theory and $SL_{2}$.




















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