I would use the new Toolbar instead of ActionBar. Checkout this tutorial: http://www.android4devs.com/2014/12/how-to-make-material-design-navigation-drawer.html
The important part is that you have to put the toolbar inside the DrawerLayout as seen here (activity_main.xml):
<?xml version="1.0" encoding="utf-8"?>
<android.support.v4.widget.DrawerLayout
    xmlns:android="http://schemas.android.com/apk/res/android"
    android:id="@+id/DrawerLayout"
    android:layout_width="match_parent"
    android:layout_height="match_parent"
    android:elevation="7dp">
    <LinearLayout
        android:layout_width="match_parent"
        android:layout_height="match_parent"
        android:orientation="vertical">
        <include
            android:id="@+id/tool_bar"
            layout="@layout/tool_bar">
        </include>
        <TextView
            android:layout_width="wrap_content"
            android:layout_height="wrap_content"
            android:text="Hello World" />
    </LinearLayout>
    <android.support.v7.widget.RecyclerView
        android:id="@+id/RecyclerView"
        android:layout_width="320dp"
        android:layout_height="match_parent"
        android:layout_gravity="left"
        android:background="#ffffff"
        android:scrollbars="vertical">
    </android.support.v7.widget.RecyclerView>
</android.support.v4.widget.DrawerLayout>
toolbar.xml:
<?xml version="1.0" encoding="utf-8"?>
<android.support.v7.widget.Toolbar xmlns:android="http://schemas.android.com/apk/res/android"
    android:layout_width="match_parent"
    android:layout_height="wrap_content"
    android:background="@color/ColorPrimary"
    android:theme="@style/ThemeOverlay.AppCompat.Dark"
    android:elevation="4dp"
    android:paddingTop="@dimen/tool_bar_top_padding">
</android.support.v7.widget.Toolbar>
To use the toolbar:
Toolbar toolbar = (Toolbar) findViewById(R.id.my_awesome_toolbar);
setSupportActionBar(toolbar);