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I suggest the following lecture notes of Bruhat:

www.math.tifr.res.in/~publ/ln/tifr14.pdf

Chapter 3 & 4 should answer most of your questions.

For example, there are statements like this :

Proposition 1(pg.19). To every analytic representation h : G −→ G′ there corresponds a map dh : U(G) → U(G′) which is a representation of algebras such that ( f ◦ h) = (dh() f ) ◦ h.

Corollary (pg.36). Let G and H be two Lie groups having g and J as their Lie algebras. If G is connected and simply connected, to every representation π of g in J, there corresponds one and only one representation f of G → H such that d f = π.

If you are interested in semisimple, connected and simply connected groups only, both give you an isomorphism between the categories. Equivalence of categories is weaker then isomorphism. Moreover, the categories of representations are both semisimple in this case, i.e., reps decompose into irreducible ones. Thus, also your second and third question can be answered affirmative.

I suggest the following lecture notes of Bruhat:

www.math.tifr.res.in/~publ/ln/tifr14.pdf

Chapter 3 & 4 should answer most of your questions.

For example, there are statements like this :

Proposition 1(pg.19). To every analytic representation h : G −→ G′ there corresponds a map dh : U(G) → U(G′) which is a representation of algebras such that ( f ◦ h) = (dh() f ) ◦ h.

Corollary (pg.36). Let G and H be two Lie groups having g and J as their Lie algebras. If G is connected and simply connected, to every representation π of g in J, there corresponds one and only one representation f of G → H such that d f = π.

If you are interested in semisimple, connected, simply connected groups only, both give you an isomorphism between the categories of complex representations. Equivalence of categories is weaker than isomorphism. Moreover, the categories of representations are both semisimple in this case, i.e., reps decompose into irreducible ones. Thus, also your second and third question can be answered affirmative in this case.

I suggest the following lecture notes of Bruhat:

www.math.tifr.res.in/~publ/ln/tifr14.pdf

Chapter 3 & 4 should answer most of your questions.

For example, there are statements like this :

Proposition 1(pg.19). To every analytic representation h : G −→ G′ there corresponds a map dh : U(G) → U(G′) which is a representation of algebras such that ( f ◦ h) = (dh() f ) ◦ h.

Corollary (pg.36). Let G and H be two Lie groups having g and J as their Lie algebras. If G is connected and simply connected, to every representation π of g in J, there corresponds one and only one representation f of G → H such that d f = π.

Both give you an isomorphism between the categories. Equivalence would be a weaker statement.

Since you are interested in reductive groups only, the categories of representations are both semisimple, i.e., reps decompose into irreducible ones.

  Thus, also your second and third question can be answered affirmative.

I suggest the following lecture notes of Bruhat:

www.math.tifr.res.in/~publ/ln/tifr14.pdf

Chapter 3 & 4 should answer most of your questions.

For example, there are statements like this :

Proposition 1(pg.19). To every analytic representation h : G −→ G′ there corresponds a map dh : U(G) → U(G′) which is a representation of algebras such that ( f ◦ h) = (dh() f ) ◦ h.

Corollary (pg.36). Let G and H be two Lie groups having g and J as their Lie algebras. If G is connected and simply connected, to every representation π of g in J, there corresponds one and only one representation f of G → H such that d f = π.

If you are interested in semisimple, connected and simply connected groups only, both give you an isomorphism between the categories. Equivalence of categories is weaker then isomorphism. Moreover, the categories of representations are both semisimple in this case, i.e., reps decompose into irreducible ones. Thus, also your second and third question can be answered affirmative.

I suggest the following lecture notes of Bruhat:

www.math.tifr.res.in/~publ/ln/tifr14.pdf

Chapter 3 & 4 should answer your three questions.

I suggest the following lecture notes of Bruhat:

www.math.tifr.res.in/~publ/ln/tifr14.pdf

Chapter 3 & 4 should answer most of your questions.

For example, there are statements like this :

Proposition 1(pg.19). To every analytic representation h : G −→ G′ there corresponds a map dh : U(G) → U(G′) which is a representation of algebras such that ( f ◦ h) = (dh() f ) ◦ h.

Corollary (pg.36). Let G and H be two Lie groups having g and J as their Lie algebras. If G is connected and simply connected, to every representation π of g in J, there corresponds one and only one representation f of G → H such that d f = π.

Both give you an isomorphism between the categories. Equivalence would be a weaker statement.

Since you are interested in reductive groups only, the categories of representations are both semisimple, i.e., reps decompose into irreducible ones.

Thus, also your second and third question can be answered affirmative.