Puestos de Trabajo: Tareas, Funciones y Responsabilidades

By Corollary 3.15 in Putcha’s book [20], we can considerS to be a closed subsemigroup of

Mn(K) for somen. The following definition from Putcha’s book relies on this predetermined

embedding ofS.

Definition 8.20. We define thedeterminant with respect toe as the map, dete :S → K given by dete(s) :=det(ese+1−e).

We have defined our determinant by using the ambient Mn(K). It is entirely conceivable

that the map would change based on our embedding ofS, but that will not affect our proof of Renner’s conjecture. Although we defined the determinant on all ofS, it only gains the familiar multiplicative property when we restrict toeS e.

Proposition 8.21. When restricted to eS e, deteis a multiplicative morphism.

Proof. It is clear that the map is already a morphism in the algebraic geometry sense, just from its definition. It remains to show that it is multiplicative, hence a true algebraic monoid morphism. Leta,b∈eS e. Then,

dete(a)dete(b)=det(a+1−e)det(b+1−e)

=det(ab+a−ae+b+1−e−eb−e+e)=det(ab+1−e)= dete(ab)

In light of the previous result and its desired property, we will only consider dete as re-

stricted toeS e.

Proposition 8.22. H= {m∈eS e|dete(m),0}

Proof. This comes from Remark 3.23 in [20].

The relative determinant is how Putcha shows in his book thatHis an algebraic group. The group, Hprovides theGLn(K) toeS e’s Mn(K). In the same vein we take the opportunity now

to define a useful analogue of the special linear group.

8.3. Putcha’sDeterminant 119

While Putcha shows in his paper ([19]) thatH = det−e1(K

∗

) is independent of the particular embedding in Mn(K), our definition of H1 is quite dependent on it. However, regardless of

which of the possibleH1we have, the following results show that any suchH1 = det−e1(1) has

the properties we will need to prove Renner’s conjecture.

Theorem 8.24. H1is a closed, reductive algebraic group in eS e.

Proof. ThatH1is a closed algebraic variety comes from its definition as the preimage,det−e1(1).

It is a normal connected subgroup ofH, by Proposition 8.22 and the multiplicativity ofdete in

Proposition 8.21 and the fact that it is the kernel ofdete. SinceH1is a connected closed normal

subgroup of the reductive groupH it follows thatH1is reductive by 14.2 of [3].

Proposition 8.25. Let u ∈H be a unipotent element. Then u∈H1. That is, dete(u)= 1. Proof. dete : H → K∗ is a morphism of algebraic groups. So then for any element, h ∈ H, dete(huhs) =dete(hu)dete(hs) preserves the Jordan decomposition. Hence,dete takes unipotent

elements to unipotent elements. But 1∈ K∗is the unique unipotent element ofK∗. So for any

unipotentu∈H,dete(u)= 1 as desired.

We will use H1 a great deal to reach our end result. First, however, we will move on and

introduce Renner’s maps and the action of K∗ that they provide. We fix a maximal torus of

eS e, call itT. The majority of what follows comes from Exercise 5 in Section 4.6 of [30].

Definition 8.26. Define the set of idempotents of corank 1, to be the set of all idempotents which lies just below e in the Adherence order. E1_(T)= _{{f ∈E(T)|e covers f}.}

For the above definition, remember thateis the identity element ofeS e.

Lemma 8.27. For any f0 ∈E(T),Πf∈E1_(T),f

0≤f f = f0

This is a useful way of creating a product of 0 out of our idempotents in E1_(T).

Proof. This comes from Proposition 3.22 b) in [30], although we wish to acknowledge the typo in the book, as the definition of the set “E1_{(f)” should contain a “f ≤e”.}

Proposition 8.28. For each f ∈ E1(T)there is a unique injective morphismαf :K → T such thatαf(0)= f ,αf(K∗)={t ∈T |t f = f t= f}0.

Proof. Exercise 5 b), sinceT is a D-monoid with zero. These maps are what we meant when we referred to Renner’s maps. Taken together they provide the action of K∗ that we will be using to prove Renner’s conjecture. This action interacts nicely with Putcha’s relative determinant.

Definition 8.29. For k∈K and m∈S we define the product k·m= Π_f∈E1_(T)αf(k)

m.

The following proposition shows that our product exhibits the properties of a group action when restricted toK∗. It also behaves like scalar multiplication in the sense that 0·m=0.

Proposition 8.30. For m,m0 ∈S , and k,k0 ∈K the following are true, (1) k·(k0_·m)=_(kk0_)·m (2) k·(mm0)= (k·m)m0 (3)1·m=m (4)0·m=0 Proof. (1)k·(k0·m)=k·Π_f∈E1_(T)αf(k0) m=Πf∈E1_(T)αf(k) Πf∈E1_(T)αf(k0) m

But by associativity of our monoid multiplication, we get, =

Πf∈E1_(T)αf(k)αf(k0)

m

And since each of ourαf is a morphism, and the images of our field elements are inT, which

is commutative

=

Πf∈E1_(T)αf(kk0)

m= (kk0)·m

(2) This is just an application of the associative law. (3) For each f ∈ E1_(T),α

f(1)= 1, the identity of our monoid. So then,

1·m=Π_f∈E1_(T)αf(1)

m= Π_f∈E1_(T)1

m=m

(4) By definition, for each f ∈E1_(T),α

f(0)= f. By Lemma 8.27, we can conclude,

0·m=Π_f∈E1_(T)αf(0)

m= Π_f∈E1_(T)f

m=(0)m= 0

We would like to show thatdeteis a homogenous morphism. That is, there exists someqso

thatdete(ka)=kqdete(a). This is a well-known property of the original determinant function.

Proposition 8.31. For each f ∈ E1_{(T), there exists some positive integer, q}

f, so that for every k∈K, dete(αf(k))=kqf.

8.4. TheConjecture 121

Proof. As it is a composition of a series of morphisms, we can see that p : x7→ dete(αf(x)) is

a polynomial,p(x). See that p(x) is nonzero and multiplicative,

p(xy)=dete(αf(xy))=dete(αf(x)αf(y))=dete(αf(x))dete(αf(y))= p(x)p(y).

It follows that, p(x) = xq _{for some q ∈}

N (by Lemma A.5). And since f < H, we know dete(f)=dete(αf(0))=0. But ifqf = 0 thendete(f)= 1 a contradiction. Thus, 1≤qf.

Corollary 8.32. There exists some positive integer, q, so that for m ∈ eS e and k ∈ K, dete(km)=kqdete(m). In fact, q = Σf∈E1_(T)qf.

Proof. Sincedeteis multiplicative, we see that, dete(km)=dete (Π_f∈E1_(T)αf(k))m = Πf∈E1_(T)dete(αf(k)) dete(m) = Πf∈E1_(T)kqfdete(m)= kqdete(m),

whereqis defined as in the statement above.

Thus,dete is homogeneous of degree≥1. Our final result before attempting the conjecture

shows us that our reductive group and our action ofK∗commute.

Proposition 8.33. For any k∈K, h∈H1, and s∈S , h(k·s)=k·hs.

Proof. This comes from the fact thatΠ_f∈E1_(T)αf(k)∈T ⊆Z(eS e) and H1⊆ eS e.

Now we have everything we need to discuss Renner’s conjecture properly. The group

H1 provides exactly the reductive group we need to show the closed conditions associated

to semi-stability and stability. Renner’s maps will allow us to make a Bruhat decomposition argument to show the nature of the semi-stable elements.